Bài tập : Đại số đại cương

d¯a’o cu’a x va` y. b) Gia’ su.’ a la` pha`ˆn tu.’ nghi.ch d¯a’o cu’a xy, ngh˜ıa la` a(xy) = (xy)a = 1. Ta co´ x 6= 0 va` y 6= 0, v`ı neˆ´u x = 0 hay y = 0 th`ı xy = 0 neˆn xy khoˆng co´ nghi.ch d¯a’o. D- a˘. t x′ = ya va` y′ = ax th`ı xx′ = y′y = 1. Khi d¯o´ x(x′x− 1) = xx′x− x = 1x− x = 0 ⇒ x′x− 1 = 0 ⇒ x′x = 1, do R khoˆng co´ u.´o.c cu’a khoˆng va` x 6= 0. Vaˆ.y x′ la` pha`ˆn tu.’ nghi.ch d¯a’o cu’a x. Tu.o.ng tu.. y′ la` pha`ˆn tu.’ nghi.ch d¯a’o cu’a y. 7. a) Gia’ su.’ ba = 1. Khi d¯o´ neˆ´u ac = 0 th`ı c = (ba)c = b(ac) = b0 = 0. Do d¯o´ a khoˆng la` u.´o.c cu’a 0 beˆn tra´i. Gia’ su.’ a = ara vo´.i r ∈ R va` a khoˆng la` u.´o.c cu’a 0 beˆn tra´i. Khi d¯o´ a(1− ra) = a− ara = 0 neˆn ta co´ 1− ra = 0 hay ra = 1. Do d¯o´ a co´ nghi.ch d¯a’o tra´i la` r. b) Neˆ´u to`ˆn ta. i c ∈ R sao cho c(1−ba) = 1 th`ı c(1−ba)b = b hay cb(1−ab) = b. Khi d¯o´ 1 = ab + (1 − ab) = acb(1 − ab) + (1 − ab) = (acb + 1)(1 − ab). Do d¯o´ 1− ab kha’ nghi.ch tra´i. 8. a) Vo´.i a ∈ R, a 6= 0, xe´t a´nh xa. fa : R −→ R : x 7→ ax. fa la` moˆ. t d¯o.n a´nh. Thaˆ. t vaˆ.y ∀x, y ∈ R, ax = ay ke´o theo a(x − y) = 0 neˆn x− y = 0 v`ı R la` va`nh khoˆng co´ u.´o.c cu’a khoˆng va` a 6= 0. Do R la` hu˜.u ha.n neˆn fa la` moˆ. t song a´nh. Vı` vaˆ. y, vo´.i a ∈ R to`ˆn ta. i e ∈ R sao cho fa(e) = ae = a. Ta chu´.ng minh e la` d¯o.n vi. cu’a R. ∀x ∈ R, a(ex − x) = (ae)x − ax = ax − ax = 0, v`ı a 6= 0 neˆn ex − x = 0 hay ex = x. Tu`. d¯o´ ea = a va` (xe − x)a = x(ea) − xa = xa − xa = 0, do d¯o´ xe− x = 0 hay xe = x. Vı` fa la` song a´nh neˆn vo´.i e ∈ R, to`ˆn ta. i a′ ∈ R sao cho fa(a′) = aa′ = e. Ta co´ a(a′a− e) = (aa′)a− ae = ea− ae = 0 neˆn a′a = e. Vaˆ.y a′ la` pha`ˆn tu.’ nghi.ch d¯a’o cu’a a. b) Gia’ su.’ a co´ nghi.ch d¯a’o tra´i la` a′, ngh˜ıa la` a′a = e. Xe´t a´nh xa. fa : R −→ R : x 7→ ax. fa la` moˆ. t d¯o.n a´nh. Thaˆ. t vaˆ. y ∀x, y ∈ R, ax = ay ke´o theo a′(ax) = a′(ay), do d¯o´ x = y. Do R la` hu˜.u ha.n neˆn fa la` moˆ. t song a´nh. Khi d¯o´ vo´.i d¯o.n vi. e cu’a R, to`ˆn ta. i a′′ ∈ R sao cho fa(a′′) = aa′′ = e, tu´.c la` a co´ nghi.ch d¯a’o pha’i la` a′′. Tu.o.ng tu.. neˆ´u a co´ nghi.ch d¯a’o pha’i th`ı a co´ nghi.ch d¯a’o tra´i neˆn a kha’ nghi.ch. 30 9. a) To`ˆn ta. i n, p ∈ N∗ so cho xn = yp = 0. Theo coˆng thu´.c nhi. thu´.c Newton: (x+ y)n+p−1 = n+p−1∑ k=0 Ckn+p−1x kyn+p−1−k = ( n−1∑ k=0 Ckn+p−1x kyn−1−k)yp − xn( n+p−1∑ k=n Ckn+p−1x k−nyn+p−1−k) = 0. Do d¯o´ x+ y la` lu˜y linh. b) Neˆ´u xn = 0 th`ı (xy)n = xnyn = 0. Do d¯o´ xy la` lu˜y linh. c) Neˆ´u xn = 0, ky´ hieˆ.u y = n−1∑ k=0 xk, ta co´ (1− x)y = y(1 − x) = 1− xn = 1. Do d¯o´ 1− x kha’ nghi.ch va` (1− x)−1 = y. 10. D- a˘. t A = {m n ∈ Q | (n, p) = 1 } . Ta co´ A 6= ∅ v`ı Z ⊂ A. ∀m1 n1 , m2 n2 ∈ A, m1 n1 − m2 n2 = m1n2 −m2n1 n1n2 ∈ A, m1 n1 . m2 n2 = m1m2 n1n2 ∈ A, v`ı (n1, p) = 1 va` (n2, p) = 1 neˆn (n1n2, p) = 1. Vaˆ.y A la` moˆ. t va`nh con cu’a Q, neˆn A la` moˆ. t mie`ˆn nguyeˆn. Go. i A la` tru.`o.ng ca´c thu.o.ng cu’a A th`ı do A ⊃ Z neˆn A ⊃ Q, ma˘. t kha´c v`ı A ⊂ Q neˆn A ⊂ Q. Vaˆ.y A = Q. 11. Cho R la` moˆ. t mie`ˆn nguyeˆn hu˜.u ha.n, gia’ su.’ R = {0, a1, a2, . . . , an}. Khi d¯o´ ca´c pha`ˆn tu.’ cu’a R∗ = {a1, a2, . . . , an} thoa’ ma˜n luaˆ. t gia’n u.´o.c. Do d¯o´ R∗ = {a1a1, a1a2, . . . , a1an}. Vı` a1 ∈ R∗ neˆn to`ˆn ta. i k sao cho a1ak = a1. D- a˘. t e = ak, vo´.i 1 ≤ i ≤ n ta co´ a1(eai) = (a1e)ai = a1ai, suy ra eai = ai hay e la` pha`ˆn tu.’ d¯o.n vi. cu’a R. Vo´.i mo. i aj ∈ R∗, R∗ = {a1aj, a2aj, . . . , anaj}. Vı` e ∈ R∗ neˆn to`ˆn ta. i ai ∈ R∗ sao cho aiaj = e hay ai la` pha`ˆn tu.’ nghi.ch d¯a’o cu’a aj. Do d¯o´ R la` moˆ. t tru.`o.ng. 12. a) ⇒ b) Cho I la` moˆ. t id¯eˆan cu’a R, I 6= {0}. Khi d¯o´ to`ˆn ta. i x ∈ I, x 6= 0, do R la` moˆ. t tru.`o.ng neˆn x kha’ nghi.ch va` ta co´ 1 = x.x−1 ∈ I. Do d¯o´ vo´.i moˆ˜i a ∈ R, a = a.1 ∈ I, neˆn I = R. b) ⇒ c) Cho f la` moˆ. t d¯o`ˆng caˆ´u va`nh kha´c khoˆng tu`.R va`o va`nh kha´c khoˆng S. Khi d¯o´ Kerf la` moˆ. t id¯eˆan cu’a R, neˆn Kerf = {0} hoa˘.c Kerf = R. Do f 6= 0 va` S 6= {0} neˆn Kerf = {0} hay f la` moˆ. t d¯o.n caˆ´u. c) ⇒ a) Vo´.i x ∈ R, x 6= 0, xe´t id¯eˆan sinh ra bo.’ i x, ngh˜ıa la` = {ax | a ∈ R}. Gia’ su.’ 6= R. Khi d¯o´ R/ la` va`nh kha´c khoˆng va` p : R −→ R/ : a 7→ a+ la` moˆ. t d¯o`ˆng caˆ´u va`nh kha´c khoˆng. Do d¯o´ p la` moˆ. t d¯o.n caˆ´u hay =Kerp = {0}. D- ie`ˆu voˆ ly´ na`y cho bieˆ´t = R. 31 Tu`. d¯o´ 1 ∈ hay to`ˆn ta. i a ∈ R sao cho ax = 1 hay x la` kha’ nghi.ch. Vaˆ.y R la` moˆ. t tru.`o.ng. 13. D- a˘. t T = { ( a b 3b a ) | a, b ∈ Q} th`ı T la` moˆ. t taˆ.p con kha´c roˆ˜ng cu’a va`nh M2(Q) ca´c ma traˆ.n vuoˆng caˆ´p 2 treˆn Q vo´.i phe´p coˆ.ng va` nhaˆn ma traˆ.n va` chu´.a ma traˆ.n d¯o.n vi. ( 1 0 0 1 ) . Ta co´ ( a b 3b a ) − ( a′ b′ 3b′ a′ ) = ( a− a′ b− b′ 3(b− b′) a− a′ ) ,( a b 3b a )( a′ b′ 3b′ a′ ) = ( aa′ + 3bb′ ab′ + ba′ 3(ba′ + ab′) 3bb′ + aa′ ) ,( a b 3b a )( a′ b′ 3b′ a′ ) = ( a′ b′ 3b′ a′ )( a b 3b a ) Vaˆ.y T la` moˆ. t va`nh con giao hoa´n cu’a M2(Q) co´ chu´.a d¯o.n vi. cu’a M2(Q). Do d¯o´ T la` moˆ. t va`nh giao hoa´n kha´c 0 co´ d¯o.n vi.. Ngoa`i ra, vo´.i ( a b 3b a ) 6= ( 0 0 0 0 ) (khi d¯o´ a2 − 3b2 6= 0), ta co´( a b 3b a )( a a2−3b2 −b a2−3b2 −3b a2−3b2 a a2−3b2 ) = ( 1 0 0 1 ) . Do d¯o´ T la` moˆ. t tru.`o.ng. Xe´t a´nh xa. f : T −→ A : ( a b 3b a ) 7→ a+ b √ 3. Ro˜ ra`ng f la` moˆ. t toa`n a´nh. f co`n la` moˆ. t d¯o.n a´nh v`ı vo´.i a + b √ 3 = a′ + b′ √ 3 th`ı a = a′ va` b = b′ tu´.c la` ( a b 3b a ) = ( a′ b′ 3b′ a′ ) . Ngoa`i ra, f( ( a b 3b a ) + ( a′ b′ 3b′ a′ ) ) = f( ( a+ a′ b+ b′ 3(b+ b′) a+ a′ ) ) = (a+ a′) + (b+ b′) √ 3 = (a+ b √ 3) + (a′ + b′ √ 3) = f( ( a b 3b a ) ) + f( ( a′ b′ 3b′ a′ ) ), 32 f( ( a b 3b a )( a′ b′ 3b′ a′ ) ) = f( ( aa′ + 3bb′ ab′ + ba′ 3(ba′ + ab′) 3bb′ + aa′ ) ) = (aa′ + 3bb′) + (ab′ + ba′) √ 3 = (a+ b √ 3)(a′ + b′ √ 3) = f( ( a b 3b a ) )f( ( a′ b′ 3b′ a′ ) ). Vaˆ.y f la` moˆ. t d¯a˘’ ng caˆ´u. 14. D- a˘. t T = { ( a b −b a ) | a, b ∈ R} th`ı T la` moˆ. t taˆ.p con kha´c roˆ˜ng cu’a va`nh M(2,R) ca´c ma traˆ.n vuoˆng caˆ´p 2 treˆn R vo´.i phe´p coˆ.ng va` nhaˆn ma traˆ.n va` chu´.a ma traˆ.n d¯o.n vi. ( 1 0 0 1 ) . Ta co´ ( a b −b a ) − ( a′ b′ −b′ a′ ) = ( a− a′ b− b′ −(b− b′) a− a′ ) ,( a b −b a )( a′ b′ −b′ a′ ) = ( aa′ − bb′ ab′ + ba′ −(ba′ + ab′) −bb′ + aa′ ) ,( a b −b a )( a′ b′ −b′ a′ ) = ( a′ b′ −b′ a′ )( a b −b a ) Vaˆ.y T la` moˆ. t va`nh con giao hoa´n cu’a M(2,R) co´ chu´.a d¯o.n vi. cu’a M(2,R). Do d¯o´ T la` moˆ. t va`nh giao hoa´n kha´c 0 co´ d¯o.n vi.. Ngoa`i ra, vo´.i ( a b −b a ) 6= ( 0 0 0 0 ) (khi d¯o´ a2 + b2 6= 0), ta co´( a b −b a )( a a2+b2 −b a2+b2 b a2+b2 a a2+b2 ) = ( 1 0 0 1 ) . Do d¯o´ T la` moˆ. t tru.`o.ng. Xe´t a´nh xa. f : T −→ A : ( a b −b a ) 7→ a+ bi. Ro˜ ra`ng f la` moˆ.t toa`n a´nh. f co`n la` moˆ. t d¯o.n a´nh v`ı vo´.i a + bi = a′ + b′i th`ı a = a′ va` b = b′ tu´.c la` ( a b −b a ) = ( a′ b′ −b′ a′ ) . Ngoa`i ra, f( ( a b −b a ) + ( a′ b′ −b′ a′ ) ) = f( ( a+ a′ b+ b′ −(b+ b′) a+ a′ ) ) = (a+ a′) + (b+ b′)i = (a+ bi) + (a′ + b′i) = f( ( a b −b a ) ) + f( ( a′ b′ −b′ a′ ) ), 33 f( ( a b −b a )( a′ b′ −b′ a′ ) ) = f( ( aa′ − bb′ ab′ + ba′ −(ba′ + ab′) −bb′ + aa′ ) ) = (aa′ − bb′) + (ab′ + ba′)i = (a+ bi)(a′ + b′i) = f( ( a b −b a ) )f( ( a′ b′ −b′ a′ ) ). Vaˆ.y f la` moˆ. t d¯a˘’ ng caˆ´u. 15. a) Ta co´ Q(√p) la` moˆ. t taˆ.p con kha´c roˆ˜ng cu’a tru.`o.ng R ca´c soˆ´ thu.. c va` co´ chu´.a soˆ´ nguyeˆn 1 (v`ı 1 = 1 + 0 √ p). ∀a, b, a′, b′ ∈ Q, (a+ b √ p)− (a′ + b′√p) = (a− a′) + (b− b′)√p, (a+ b √ p)(a′ + b′ √ p) = (aa′ + pbb′) + (ab′ + ba′) √ p. Vaˆ.y Q( √ p) la` moˆ. t va`nh con cu’a R chu´.a 1, neˆn no´ la` moˆ. t va`nh giao hoa´n co´ d¯o.n vi.. Ngoa`i ra, vo´.i a+ b √ p ∈ Q(√p) va` kha´c 0 (a va` b khoˆng d¯o`ˆng tho`.i ba`˘ng 0), ta co´ a2− pb2 6= 0, aa2−pb2 + −ba2−pb2 ∈ Q( √ p) va` (a+ b √ p)( aa2−pb2 + −b a2−pb2 ) = 1. Do d¯o´ Q(√p) la` moˆ. t tru.`o.ng. b) Gia’ su.’ to`ˆn ta. i d¯a˘’ ng caˆ´u tru.`o.ng f : Q( √ 7) −→ Q(√11). Khi d¯o´ f(1) 6= 0 va` do f(1) = f(1.1) = f(1)f(1) neˆn f(1) = 1. Tu`. d¯o´ f(7) = f(7.1) = 7f(1) = 7. Gia’ su.’ f( √ 7) = a+ b √ 11 (vo´.i a, b ∈ Q). Ta co´ 7 = f(7) = f( √ 7. √ 7) = f( √ 7)2 = (a+ b √ 11)2 hay a2 + 11b2 + 2ab √ 11 = 7 hay 2ab √ 11 = 7− a2 − 11b2. – Neˆ´u a = b = 0 th`ı 0 = 7: voˆ ly´. – Neˆ´u a = 0 va` b 6= 0 th`ı b = √ 7 11 : voˆ ly´. – Neˆ´u b = 0 va` a 6= 0 th`ı a = √7: voˆ ly´. – Neˆ´u a 6= 0 va` b 6= 0 th`ı √11 = 7−a2−11b2 2ab : voˆ ly´ v`ı veˆ´ pha’i la` moˆ. t soˆ´ hu˜.u tı’ nhu.ng veˆ´ tra´i la` moˆ. t soˆ´ voˆ tı’. 16. Gia’ su.’ f : F −→ F la` moˆ. t tu.. d¯o`ˆng caˆ´u cu’a tru.`o.ng F. Khi d¯o´ f(1)(f(1)− 1) = f(1)f(1)− f(1) = f(1.1)− f(1) = 0, do d¯o´ f(1) = 0 hay f(1) = 1. – Neˆ´u f(1) = 0 th`ı f(a) = f(a.1) = f(a)f(1) = f(a).0 = 0, ∀a ∈ F, neˆn ta co´ f = 0. – Neˆ´u f(1) = 1 th`ı ta la`ˆn lu.o.. t xe´t F la` Q, R, Zp, C. O .’ d¯aˆy, f la` d¯o.n caˆ´u v`ı vo´.i x 6= 0, ta co´ f(x)f(x−1) = f(xx−1) = f(1) = 1 6= 0 do d¯o´ f(x) 6= 0. a) ∀n ∈ Z, f(n) = f(n.1) = nf(1) = n.1 = n. 34 ∀q ∈ Q, q = n m , n,m ∈ Z, m 6= 0, ta co´ mf(q) = f(mq) = f(n) = n, do d¯o´ f(q) = n m = q. Vaˆ.y ca´c tu.. d¯o`ˆng caˆ´u cu’a tru.`o.ng Q la` a´nh xa. 0 va` a´nh xa. d¯o`ˆng nhaˆ´t. b) Tru.´o.c heˆ´t, neˆ´u r ∈ R, r > 0 th`ı f(r) > 0. Thaˆ.t vaˆ.y, f(r) = f( √ r. √ r) = f( √ r)2 > 0 (v`ı neˆ´u f( √ r) = 0 th`ı do f d¯o.n caˆ´u ta co´ √ r = 0 hay r = 0). f la` ha`m ta˘ng. Thaˆ. t vaˆ. y, ∀x, y ∈ R, x 0, neˆn f(y)−f(x) = f(y − x) > 0 hay f(x) < f(y). Gia’ su.’ ∃z ∈ R sao cho f(z) 6= z. Neˆ´u f(z) < z th`ı tu`. t´ınh tru` maˆ. t cu’a Q trong R to`ˆn ta. i q ∈ Q sao cho f(z) < q < z. Khi d¯o´ q = f(q) < f(z), maˆu thuaˆ˜n vo´.i f(z) z cu˜ng daˆ˜n d¯eˆ´n maˆu thuaˆ˜n. Do d¯o´ f(z) = z, ∀z ∈ R. Vaˆ.y ca´c tu.. d¯o`ˆng caˆ´u cu’a tru.`o.ng R la` a´nh xa. 0 va` a´nh xa. d¯o`ˆng nhaˆ´t. c) ∀a ∈ Zp, f(a) = f(a.1) = af(1) = a.1 = a. Vaˆ.y ca´c tu.. d¯o`ˆng caˆ´u cu’a tru.`o.ng Zp la` a´nh xa. 0 va` a´nh xa. d¯o`ˆng nhaˆ´t. d) Gia’ su.’ f : C −→ C la` moˆ. t tu.. d¯o`ˆng caˆ´u cu’a tru.`o.ng soˆ´ phu´.c C sao cho f(a) = a vo´.i mo. i a ∈ R. Nhu. vaˆ.y vo´.i soˆ´ phu´.c baˆ´t ky` z = a + ib ∈ C, ta co´ f(z) = f(a+ ib) = f(a) + f(b)f(i) = a+ bf(i). Vı` i2 = −1 neˆn [f(i)]2 = f(i2) = f(−1) = −1. Do d¯o´ f(i) = i hoa˘.c f(i) = −i. Vaˆ. y ca´c tu.. d¯o`ˆng caˆ´u cu’a tru.`o.ng C la` a´nh xa. d¯o`ˆng nhaˆ´t va` a´nh xa. cho lieˆn ho.. p (z 7→ z). 17. a) Ro˜ ra`ng ( 1 0 0 1 ) ∈ Q. ∀A = ( a b −b a ) , A′ = ( a′ b′ −b′ a′ ) ∈ Q, A −A′ = ( a− a′ b − b′ −b+ b′ a− a′ ) = ( a− a′ b− b′ −(b− b′) a− a′ ) ∈ Q, AA′ = ( aa′ − bb′ ab′ + ba′ −ba′ − ab′ −bb′ + aa′ ) = ( aa′ − bb′ ab′ + ba′ −(ab′ + ba′) aa′ − bb′ ) ∈ Q. Cho A = ( a b −b a ) 6= ( 0 0 0 0 ) , tu´.c la` aa + bb > 0. Khi d¯o´ A co´ nghi.ch d¯a’o la` A−1 =  a aa+ bb −b aa+ bb b aa+ bb a aa+ bb  ∈ Q b) De˜ˆ da`ng kieˆ’m tra d¯u.o.. c I2 = J2 = K2 = −1, IJ = −JI = K, JK = −KJ = I, KI = −IK = J . 35 ∀A = ( a b −b a ) ∈ Q, d¯a˘. t a = a1+ ia2, b = b1 + ib2, a1, a2, b1, b2 ∈ R, ta co´ A = ( a1 + ia2 b1 + ib2 −b1 + ib2 a1 − ia2 ) = ( a1 0 0 a1 ) + ( ia2 0 0 −ia2 ) + ( 0 b1 −b1 0 ) + ( 0 ib2 ib2 0 ) = a1 + a2I + b1J + b2K. 18. xy = x(x−1 + y−1)y = y + x = −1. 1 = (x+ y)2 = x2 + 2xy + y2 ⇒ x2 + y2 = 1− 2xy = 3. 9 = (x2 + y2)2 = x4 + 2x2y2 + y4 ⇒ x4 + y4 = 9− 2(xy)2 = 9− 2 = 7. 19. Gia’ su.’ to`ˆn ta. i moˆ. t d¯a˘’ ng caˆ´u nho´m f : (K,+) −→ (K \ {0}, .). 1) 1 + 1 = 0 (1 va` 0 la`ˆn lu.o.. t la` d¯o.n vi. va` pha`ˆn tu.’ khoˆng cu’a K). ∀x ∈ K, x+ x = x(1 + 1) = x.0 = 0, (f(x))2 = f(x+ x) = f(0) = 1, f(x) = 1 hoa˘.c f(x) = −1 = 1. Nhu. vaˆ.y f(K) = {1}, K hu˜.u ha.n. D- ie`ˆu na`y voˆ ly´ v`ı K va` K \ {0} khoˆng co´ cu`ng soˆ´ pha`ˆn tu.’ . 2) 1 + 1 6= 0. Ta ky´ hieˆ.u α = f−1(1), β = f−1(−1). Ta co´{ f(2α) = f(α+ α) = (f(α))2 = 12 = 1, f(2β) = f(β + β) = (f(β))2 = (−1)2 = 1. Tu`. d¯o´ 2α = 2β hay (1 + 1)(α− β) = 0 v`ı f la` song a´nh. D- ie`ˆu na`y voˆ ly´ v`ı 1 + 1 6= 0 va` α 6= β. 20. I va` J kha´c roˆ˜ng v`ı chu´.a ma traˆ.n khoˆng. Ta co´ 0 0 0a 0 0 b 2c 0 −  0 0 0a′ 0 0 b′ 2c′ 0  =  0 0 0a− a′ 0 0 b− b′ 2(c − c′) 0  ,  x 0 0y z 0 t u v  0 0 0a 0 0 b 2c 0  =  0 0 0za 0 0 ua+ vb 2vc 0  ,  0 0 0a 0 0 b 2c 0 x 0 0y z 0 t u v  =  0 0 0ax 0 0 bx+ 2cy 2cz 0  . 36 Do d¯o´ I la` moˆ. t id¯eˆan hai ph´ıa cu’a T . 0 0 0l 0 0 2m 2n 0 −  0 0 0l′ 0 0 2m′ 2n′ 0  =  0 0 0l − l′ 0 0 2(m −m′) 2(n− n′) 0  ,  0 0 0a 0 0 b 2c 0  0 0 0l 0 0 2m 2n 0  =  0 0 00 0 0 2cl 0 0  ,  0 0 0l 0 0 2m 2n 0  0 0 0a 0 0 b 2c 0  =  0 0 00 0 0 2na 0 0  . Do d¯o´ J la` moˆ. t id¯eˆan hai ph´ıa cu’a I. 0 0 0l 0 0 2m 2n 0 x 0 0y z 0 t u v  =  0 0 0lx 0 0 2(mx+ ny) 2nz 0  , x 0 0y z 0 t 1 v  0 0 01 0 0 2m 2n 0  =  0 0 0z 0 0 1 + 2vm 2vn 0  . Do d¯o´ J la` moˆ. t id¯eˆan pha’i cu’a T nhu.ng khoˆng la` moˆ. t id¯eˆan tra´i cu’a T . 21. a) Ta chu´.ng minh ca´c id¯eˆan cu’a Z la` nZ = {nx | x ∈ Z} vo´.i n ∈ N tuy` y´. Cho I la` moˆ. t id¯eˆan cu’a Z. Neˆ´u I = {0} th`ı I = nZ vo´.i n = 0. Neˆ´u I 6= {0} th`ı go. i n la` soˆ´ nguyeˆn du.o.ng nho’ nhaˆ´t sao cho n ∈ I, ta co´ I = nZ. D- a’o la. i, vo´.i n la` moˆ. t soˆ´ tu.. nhieˆn tuy` y´ th`ı kieˆ’m tra d¯u.o.. c nZ la` moˆ. t id¯eˆan cu’a Z. b) Cho da˜y ta˘ng ca´c id¯eˆan cu’a Z: I1 ⊂ I2 ⊂ ã ã ã ⊂ In ⊂ ã ã ã Khi d¯o´ to`ˆn ta. i ca´c soˆ´ tu.. nhieˆn k1, k2, . . . , kn . . . sao cho Ij = kjZ, vo´.i j ≥ 1. Do mZ ⊂ nZ khi va` chı’ khi n∣∣m, neˆn ta co´ k2∣∣k1, k3∣∣k2, . . . , kn+1∣∣kn, . . . . Vı` vaˆ. y, to`ˆn ta. i i sao cho ki = ki+1 = ki+2 = ã ã ã , tu´.c la` Ij = Ii, ∀j > i. Da˜y gia’m ca´c id¯eˆan cu’a Z: pZ ⊃ p2Z ⊃ ã ã ã ⊃ pnZ ⊃ ã ã ã 37 la` khoˆng du`.ng. 22. Ta co´ ca´c toa`n caˆ´u va`nh sau: p1 : Rì S −→ R : (x, y) 7→ x, p2 : R ì S −→ S : (x, y) 7→ y. Neˆ´u I la` moˆ. t id¯eˆan cu’a R va` J la` moˆ. t id¯eˆan cu’a S th`ı de˜ˆ da`ng co´ d¯u.o.. c I ì J la` moˆ. t id¯eˆan cu’a R ì S. Cho M la` moˆ. t id¯eˆan cu’a va`nh t´ıch R ì S. D- a˘. t I = p1(M) va` J = p2(M) th`ı I va` J la`ˆn lu.o.. t la` id¯eˆan cu’a R va` S. ∀(x, y) ∈ M, x = p1(x, y) ∈ I, y = p2(x, y) ∈ J neˆn (x, y) ∈ I ì J . D- a’o la. i, ∀(x, y) ∈ I ì J, ∃x1 ∈ R, y1 ∈ S sao cho (x, y1), (x1, y) ∈M ; khi d¯o´ (x, y) = (x, 0) + (0, y) = (1R, 0)(x, y1) + (0, 1S)(x1, y) ∈M, trong d¯o´ 1R va` 1S la`ˆn lu.o.. t la` d¯o.n vi. cu’a R va` S. Do d¯o´ M = I ì J . Ca´c id¯eˆan cu’a va`nh va`nh Z2 la` nZìmZ trong d¯o´ n,m ∈ N. Ca´c id¯eˆan cu’a va`nh R2 la` {(0, 0)}, {0} ì R, Rì {0} va` Rì R. 23. a) Cho I la` id¯eˆan cu.. c d¯a. i, gia’ su.’ xy ∈ I x /∈ I ⇒ I + (x) = R ⇒ 1 = h+ rx, h ∈ I, r ∈ R⇒ y = hy + rxy ∈ I ⇒ I la` id¯eˆan nguyeˆn toˆ´. b) Cho I la` id¯eˆan nguyeˆn toˆ´ va` J la` id¯eˆan sao cho I ⊂ 6= J . Khi d¯o´ ∃x ∈ J, x /∈ I ⇒ ∃n > 1, xn = x⇒ x(xn−1− 1) = 0 ∈ I x/∈I⇒ z = xn−1− 1 ∈ I ⊂ J ⇒ 1 = xn−1 − z ∈ J ⇒ J = R. Vaˆ.y I la` id¯eˆan cu.. c d¯a. i. 24. a) Ro˜ ra`ng Z[i] 6= ∅. ∀a+ ib, c+ id ∈ Z[i], ta co´ (a+ ib)− (c + id) = (a− c) + i(b− d) ∈ Z[i], (a+ ib)(c+ id) = (ac− bd) + i(ad+ bc) ∈ Z[i]. Do d¯o´ Z[i] la` moˆ. t va`nh con cu’a C. Ro˜ ra`ng 6= ∅. ∀ux, uy ∈, ∀z ∈ Z[i], ta co´ ux − uy = u(x − y) ∈, z(ux) = u(zx) ∈. Vaˆ. y la` id¯eˆan cu’a Z[i]. b) Z[i]/ co´ 4 pha`ˆn tu.’ 0, 1, i, 1 + i, trong d¯o´, x = x+ , vo´.i x ∈ Z[i]. Nhu. vaˆ. y, 1 + i 6= 0 va` 1 + i 1 + i = 2i = 0. Do d¯o´ Z[i]/ co´ u.´o.c cu’a khoˆng, neˆn no´ khoˆng la` moˆ. t tru.`o.ng. c) Z[i]/ co´ 9 pha`ˆn tu.’ α0 = 0, α1 = 1, α2 = 2, α3 = i, α4 = 1 + i, α5 = 2 + i, α6 = 2i, α7 = 1 + 2i, α8 = 2 + 2i. 38 Z[i]/ la` va`nh giao hoa´n co´ d¯o.n vi. 1 = α1. Ngoa`i ra, α−11 = α1, α −1 2 = α2, α −1 3 = α6, α −1 4 = α5, α−15 = α4, α −1 6 = α3, α −1 7 = α8, α −1 8 = α7. Vaˆ.y Z[i]/ la` moˆ. t tru.`o.ng. 25. Ro˜ ra`ng I = {ra | r ∈ R} 6= ∅ va` J = {ra + na | r ∈ R va` n ∈ Z} 6= ∅. ∀r, s, t ∈ R, ∀n,m ∈ Z, ra− sa = (r − s)a ∈ I, t(ra) = (tr)a ∈ I, (ra+ na)− (sa+ma) = (r − s)a+ (n −m)a ∈ J , t(ra + na) = (tr + nt)a+ 0a ∈ J . Vaˆ.y {ra | r ∈ R} va` {ra+ na | r ∈ R va` n ∈ Z} la` ca´c id¯eˆan cu’a R. Neˆ´u R co´ d¯o.n vi. 1 th`ı a = 1a ∈ J = {ra | r ∈ R}. Gia’ su.’ J la` moˆ. t id¯eˆan cu’a R chu´.a a. Khi d¯o´ ra ∈ J, ∀r ∈ R hay {ra | r ∈ R} ⊂ J . Do d¯o´ {ra | r ∈ R} la` id¯eˆan nho’ nhaˆ´t cu’a R chu´.a a. Vaˆ. y I = {ra | r ∈ R}. Neˆ´u R khoˆng co´ d¯o.n vi. th`ı ta co´ a = 0a+ 1a ∈ {ra+ na | r ∈ R va` n ∈ Z}. Gia’ su.’ K la` moˆ. t id¯eˆan cu’a R chu´.a a. Khi d¯o´ ra + na ∈ K, ∀r ∈ R, ∀n ∈ Z. Khi d¯o´ {ra + na | r ∈ R va` n ∈ Z} ⊂ K. Do d¯o´ {ra + na | r ∈ R va` n ∈ Z} la` id¯eˆan nho’ nhaˆ´t cu’a R chu´.a a. Vaˆ. y I = {ra+ na | r ∈ R va` n ∈ Z}. 26. a) Ro˜ ra`ng M(2,F) 6= ∅. Ky´ hieˆ.u I = {( a b 0 0 ) ∈ M(2,F) | a, b ∈ F } . Vo´.i mo. i ( a b 0 0 ) , ( a′ b′ 0 0 ) ∈ I, ( x y z t ) ∈M(2,F),( a b 0 0 ) − ( a′ b′ 0 0 ) = ( a− a′ b− b′ 0 0 ) ∈ I,( a b 0 0 )( x y z t ) = ( ax+ bz ay + bt 0 0 ) ∈ I,( x y z t )( a b 0 0 ) = ( xa xb za zb ) /∈ I khi z 6= 0 va` a 6= 0. Vaˆ.y I la` 1 id¯eˆan pha’i ma` khoˆng la` id¯eˆan tra´i cu’a M(2,F). 39 b) Cho I la` id¯eˆan kha´c khoˆng cu’a M(2,Z2). Laˆ´y A = (aij) ∈ I, A 6= 0, neˆn to`ˆn ta. i ars 6= 0. Go. i Iij ∈M(2,Z2) la` ma traˆ.n ma` pha`ˆn tu.’ do`ng i, coˆ. t j ba`˘ng 1 va` ca´c pha`ˆn tu.’ co`n la. i ba˘`ng 0. Ta co´ Isr.A.Iss = Iss, v`ı A ∈ I neˆn Iss ∈ I. Vo´.i moˆ˜i i = 1, 2, IisIssIsi = Iii va` do Iss ∈ I neˆn Iii ∈ I, vo´.i mo. i i = 1, 2. Tu`. d¯o´ ma traˆ.n d¯o.n vi. I2 = I11 + I22 ∈ I va` d¯ie`ˆu na`y daˆ˜n d¯eˆ´n I = M(2,Z2). 27. Laˆ´y a ∈ R \ {0}. Nho´m con I cu’a nho´m coˆ.ng R sinh bo.’ i a la` moˆ. t id¯eˆan cu’a R. Do d¯o´ vo´.i moˆ˜i x ∈ R, to`ˆn ta. i zx ∈ Z sao cho ax = zxa. * ∀m ∈ Z \ {0}, ma 6= 0: ∀x, y ∈ R, ∃zx, zy ∈ Z, ax = zxa, ay = zya; khi d¯o´, x = y ⇔ ax = ay ⇔ zxa = zya⇔ (zx − zy)a = 0 ⇔ zx − zy = 0 ⇔ zx = zy; do d¯o´ a´nh xa. f : R −→ Z : x 7→ zx la` moˆ. t d¯o.n a´nh. Ngoa`i ra, do a(x+y) = ax+ay = (zx+zy)a va` a(xy) = (ax)y = (zxa)y = zx(ay) = zx(zya) = (zxzy)a neˆn f(x + y) = f(x) + f(y), f(xy) = f(x)f(y). Vaˆ.y f la` moˆ. t d¯o.n caˆ´u va`nh hay R d¯a˘’ ng caˆ´u vo´.i va`nh con f(R) cu’a va`nh Z. * ∃m ∈ Z\{0}, ma = 0: Go. i p la` soˆ´ nguyeˆn du.o.ng nho’ nhaˆ´t sao cho pa = 0. Neˆ´u p = qr vo´.i 1 < q, r < p th`ı qa.ra = qra2 = pa.a = 0 neˆn qa = 0 hay ra = 0 (do R khoˆng co´ u.´o.c cu’a khoˆng). D- ie`ˆu na`y daˆ˜n d¯eˆ´n maˆu thuaˆ’n vo´.i t´ınh nho’ nhaˆ´t cu’a p. Vaˆ. y p la` moˆ. t soˆ´ nguyeˆn toˆ´. ∀x, y ∈ R, ∃zx, zy ∈ Z, ax = zxa, ay = zya; khi d¯o´, x = y ⇔ ax = ay ⇔ zxa = zya⇔ (zx−zy)a = 0 ⇔ p|(zx−zy) ⇔ zx = zy (trong Zp); do d¯o´ a´nh xa. f : R −→ Zp : x 7→ zx la` moˆ. t d¯o.n a´nh. Do zx + zy = zx + zy, zxzy = zxzy neˆn f la` moˆ. t d¯o.n caˆ´u va`nh. Vı` f 6= 0 neˆn f(R) = Zp hay f co`n la` moˆ. t toa`n caˆ´u. Vaˆ. y f la` moˆ. t d¯a˘’ ng caˆ´u va`nh. 28. a) Ma traˆ.n A la` u.´o.c beˆn tra´i cu’a khoˆng trong va`nh M khi va` chı’ khi to`ˆn ta. i B ∈ M sao cho B 6= 0 va` AB = 0, tu´.c la` khi va` chı’ khi to`ˆn ta. i X ∈ Rn sao cho X 6= 0 va` AX = 0 (ch´ınh la` moˆ.t coˆ. t cu’a B). D- ie`ˆu na`y tu.o.ng d¯u.o.ng vo´.i det(A) = 0. Tu.o.ng tu.. cho u.´o.c beˆn pha’i cu’a khoˆng ba`˘ng ca´ch thay coˆ. t bo.’ i do`ng. 40 b) Ro˜ ra`ng N 6= ∅; ngoa`i ra, a11 a12 . . . a1n 0 0 . . . 0 ... ... . . . ... 0 0 . . . 0 −  b11 b12 . . . b1n 0 0 . . . 0 ... ... . . . ... 0 0 . . . 0  =  a11 − b11 a12 − b12 . . . a1n − b1n 0 0 . . . 0 ... ... . . . ... 0 0 . . . 0  ,  a11 a12 . . . a1n 0 0 . . . 0 ... ... . . . ... 0 0 . . . 0  .  b11 b12 . . . b1n 0 0 . . . 0 ... ... . . . ... 0 0 . . . 0  =  a11b11 a11b12 . . . a11b1n 0 0 . . . 0 ... ... . . . ... 0 0 . . . 0  . Do d¯o´ N la` moˆ. t va`nh con cu’a M va` mo. i ma traˆ.n kha´c khoˆng cu’a N d¯e`ˆu la` u.´o.c beˆn pha’i cu’a khoˆng trong va`nh N . Thaˆ.t vaˆ.y, trong t´ıch 2 ma traˆ.n o.’ treˆn, cho.n a11 = 0 va` a12, . . . , a1n khoˆng d¯o`ˆng tho`.i ba˘`ng 0 th`ı t´ıch na`y ba˘`ng ma traˆ.n khoˆng. Ma traˆ.n trong N ma` pha`ˆn tu.’ do`ng 1 coˆ. t 1 kha´c 0 (a11 6= 0) d¯e`ˆu khoˆng pha’i la` u.´o.c beˆn tra´i cu’a khoˆng trong va`nh N . c) Ca´c d¯o.n vi. tra´i trong N la` ca´c ma traˆ.n co´ pha`ˆn tu.’ do`ng 1 coˆ.t 1 ba˘`ng 1 va` ca´c pha`ˆn tu.’ kha´c cu’a do`ng 1 la` tuy` y´. 29. a) Moˆ˜i d¯o`ˆng caˆ´u nho´m coˆ.ng f : Zm −→ Zn xa´c d¯i.nh gia´ tri. a = f(1) ∈ Zn va` do ma = mf(1) = f(m1) = f(m) = f(0) = 0 neˆn ta co´ caˆ´p cu’a a trong Zn la` moˆ. t u.´o.c cu’a m (do d¯o´ la` moˆ. t u.´o.c chung cu’a m va` n). D- a’o la. i, pha`ˆn tu.’ a ∈ Zn co´ caˆ´p la` moˆ. t u.´o.c cu’a m th`ı phe´p tu.o.ng u´.ng f : Zm −→ Zn : k 7→ ka la` moˆ. t a´nh xa. va` khi d¯o´ f la` moˆ. t d¯o`ˆng caˆ´u nho´m coˆ.ng. (72, 30) = 6 va` ca´c pha`ˆn tu.’ cu’a Z42 co´ caˆ´p u.´o.c cu’a 6 la`: 0, 5, 10, 15, 20, 25. Do d¯o´ ca´c d¯o`ˆng caˆ´u nho´m coˆ.ng tu`. Z72 va`o Z30 la`: f0(1) = 0, f1(1) = 5, f2(1) = 10, f3(1) = 15, f4(1) = 20, f5(1) = 25. D- o`ˆng caˆ´u nho´m coˆ.ng f : Z72 −→ Z30 la` d¯o`ˆng caˆ´u va`nh ⇔ ∀x, y ∈ Z72, f(x y) = f(x)f(y) ⇔ f(1)2 = f(1). Vaˆ.y taˆ´t ca’ ca´c d¯o`ˆng caˆ´u va`nh tu`. Z72 va`o Z30 la`: f0(k) = 0, f2(k) = 10k, f3(k) = 15k, f5(k) = 25k. 41 b) Imf0 = 0Z30 = {0}, Kerf0 = Z72. Imf2 = 10Z30 = {0, 10, 20}, Kerf2 = 3Z72, Imf3 = 15Z30 = {0, 15}, Kerf3 = 2Z72, Imf5 = 25Z30 = {0, 25, 20, 15, 10, 35}, Kerf5 = 6Z72. 30. a) ∀r, r′ ∈ R, φ(r+r′) = (r+r′+I, r+r′+J) = (r+I, r+J)+(r′+I, r′+J) = φ(r) + φ(r′) va` φ(rr′) = (rr′ + I, rr′ + J) = ((r + I)(r′ + I), (r + J)(r′ + J)) = (r + I, r + J)(r′ + I, r′ + J) = φ(r)φ(r′). Do d¯o´ φ la` moˆ. t d¯o`ˆng caˆ´u va`nh. r ∈ Kerφ⇔ φ(r) = (r+ I, r+J) = (I, J) ⇔ r+ I = I va` r+J = J ⇔ r ∈ I va` r ∈ J ⇔ r ∈ I ∩ J . Do d¯o´ Kerφ = I ∩ J . b) Do I + J = R, to`ˆn ta. i a ∈ I, b ∈ J sao cho a+ b = 1. ∀(s+ I, t+ J) ∈ R/I ìR/J, ∃r = sb+ ta (o.’ d¯aˆy s = sa+ sb va` t = ta+ tb) sao cho r − s = ta − sa = (t − s)a ∈ I va` r − t = (s − t)b ∈ J , tu´.c la` φ(r) = (r + I, r + J) = (s+ I, t+ J). Do d¯o´ φ la` moˆ. t toa`n caˆ´u va`nh. Vı` IJ ⊂ I va` IJ ⊂ J neˆn IJ ⊂ I ∩ J . Vo´.i mo. i r ∈ I ∩ J , r = ar + br ∈ IJ hay I ∩ J ⊂ IJ . Vaˆ. y I ∩ J = IJ . φ la` moˆ. t toa`n caˆ´u va` Kerφ = IJ , neˆn ta co´ d¯a˘’ ng caˆ´u: R/IJ ∼= R/I ìR/J. 31. a) Do [x3 + x] = [0] neˆn [x3] = −[x] = [x], [x4] = [x2]. Z2[x]/(x3 + x) = {[0], [1], [x], [x+ 1], [x2], [x2 + 1], [x2 + x], [x2 + x+ 1]}. . [0] [1] [x] [x+ 1] [x2] [x2 + 1] [x2 + x] [x2 + x+ 1] [0] [0] [0] [0] [0] [0] [0] [0] [0] [1] [0] [1] [x] [x+ 1] [x2] [x2 + 1] [x2 + x] [x2 + x+ 1] [x] [0] [x] [x2] [x2 + x] [x] [0] [x2 + x] [x2] [x+ 1] [0] [x+ 1] [x2 + x] [x2 + 1] [x2 + x] [x2 + 1] [0] [x+ 1] [x2] [0] [x2] [x] [x2 + x] [x2] [0] [x2 + x] [x] [x2 + 1] [0] [x2 + 1] [0] [x2 + 1] [0] [x2 + 1] [0] [x2 + 1] [x2 + x] [0] [x2 + x] [x2 + x] [0] [x2 + x] [0] [0] [x2 + x] [x2 + x [0] [x2 + x [x2] [x+ 1] [x] [x2 + 1] [x2 + x] [0] +1] +1] b) Do [x][x2+1] = [0], [x+1][x2+x] = [0], [x2][x2+1] = [0] va` [x2+x+1]2 = 1, S chı’ co´ hai pha`ˆn tu.’ kha’ nghi.ch la` ]1] va` [x2 + x+ 1]. 42 32. x2 − x + 1 la` moˆ. t d¯a thu´.c baˆ.c hai co´ ∆ = −3 < 0 neˆn khoˆng co´ nghieˆ.m trong R, do d¯o´ no´ baˆ´t kha’ quy trong R[x]. Va`nh R[x] la` mie`ˆn nguyeˆn ca´c id¯eˆan ch´ınh, ngh˜ıa la` neˆ´u I la` moˆ. t id¯eˆan cu’a R[x] th`ı I sinh ra bo.’ i moˆ. t d¯a thu´.c f(x) ∈ R[x] na`o d¯o´. Cho p(x) la` moˆ. t d¯a thu´.c baˆ´t kha’ quy trong R[x] va` J la` moˆ. t id¯eˆan cu’a R[x] sao cho (p(x)) ⊂ 6= J ⊂ R[x]. Khi d¯o´ J = (g(x)) vo´.i g(x) ∈ R[x], g(x) 6= 0, g(x)|p(x). Do p(x) baˆ´t kha’ quy neˆn g(x) = c (ha˘`ng soˆ´ kha´c 0), suy ra 1 = 1 c .c ∈ J hay J = R[x]. Vaˆ.y (p(x)) la` id¯eˆan cu.. c d¯a. i. 33. Ta co´ a4 = −1. Do 0 = 0 + 0.a+ 0.a2 + 0.a3 ∈ S neˆn S 6= ∅. ∀m,n, p, q,m′, n′, p′, q′ ∈ Z, (m+na+pa2+qa3)−(m′+n′a+p′a2+q′a3) = (m−m′)+(n−n′)a+(p−p′)a2+(q−q′)a3 ∈ S, (m+na+pa2+qa3)(m′+n′a+ p′a2 + q′a3) = (mm′ −nq′− qn′− pp′)+ (mn′ +nm′− pq′− qp′)a+(mp′ + pm′ + nn′ − qq′)a2 + (mq′ + qm′ + np′ + pn′)a3 ∈ S va` a = 0 + 1.a+ 0.a2 + 0.a3 ∈ S. Do d¯o´ S la` moˆ. t va`nh con cu’a C chu´.a a. Gia’ su.’ T la` moˆ. t va`nh con cu’a C chu´.a a. Khi d¯o´ ∀m,n, p, q ∈ Z, m = (−m)a4, na, pa2, qa3 ∈ T neˆn m + na + pa2 + qa3 ∈ T . Do d¯o´ S ⊂ T . Vaˆ. y S la` va`nh con nho’ nhaˆ´t cu’a C chu´.a a hay S la` va`nh con sinh bo.’ i a. S khoˆng la` moˆ. t id¯eˆan cu’a C v`ı vo´.i i √ 2 2 ∈ C, 1 ∈ S, d¯a˘’ ng thu´.c i √ 2 2 = i √ 2 2 .1 = m+ na+ pa2 + qa3 = ( m+ n √ 2 2 − q √ 2 2 ) + (n√2 2 + q √ 2 2 + p ) i khoˆng xa’y ra vo´.i mo. i m,n, p, q ∈ Z. 34. a) Neˆ´u n = rs, trong d¯o´ 0 < r, s < n th`ı 0 = n.1 = (rs).1 = (r.1)(s.1) va` do D la` moˆ. t mie`ˆn nguyeˆn neˆn suy ra r.1 = 0 hoa˘.c s.1 = 0. D- ie`ˆu na`y voˆ ly´ v`ı n la` caˆ´p cu’a 1. Do d¯o´ n la` moˆ. t soˆ´ nguyeˆn toˆ´. b) Do n la` moˆ.t soˆ´ nguyeˆn toˆ´ neˆn ca´c heˆ. soˆ´ nhi. thu´.c Newton Cin = n! i!(n− i)! la` boˆ. i soˆ´ cu’a n vo´.i mo. i i thoa’ ma˜n 0 < i < p. Tu`. d¯o´ v`ı D giao hoa´n, ta co´: ϕ(x + y) = xn + yn = ϕ(x) + ϕ(y), ϕ(xy) = (xy)n = xnyn = ϕ(x)ϕ(y). Do d¯o´ ϕ la` moˆ. t d¯o`ˆng caˆ´u va`nh. 43 BA`I TAˆ. P CHU . O . NG III – MOˆD- UN 1. Ky´ hieˆ.u M = {(x, y) ∈ R ì R | y > 0}. Chu´.ng minh ra˘`ng M la` moˆ. t moˆd¯un treˆn va`nh ca´c soˆ´ thu.. c R vo´.i hai phe´p toa´n sau: ∀(x, y), (u, v) ∈M, ∀α ∈ R, (x, y) + (u, v) = (x+ u, yv), α(x, y) = (αx, yα). 2. Xe´t moˆd¯un treˆn va`nh R ca´c soˆ´ thu.. c Mn(R) go`ˆm ca´c ma traˆ.n vuoˆng caˆ´p n heˆ. soˆ´ thu.. c. Ky´ hieˆ.u S(n) la` taˆ.p ho.. p ca´c ma traˆ.n d¯oˆ´i xu´.ng thu.. c caˆ´p n (tu´.c la` A = (aij) ∈Mn(R) ma` aij = aji) va` A(n) la` taˆ.p ho.. p ca´c ma traˆ.n pha’n d¯oˆ´i xu´.ng thu.. c caˆ´p n (tu´.c la` A = (aij) ∈Mn(R) ma` aij = −aji). Chu´.ng minh ra˘`ng: a) S(n) va` A(n) la` ca´c moˆd¯un con cu’a Mn(R). b) Mn(R) = S(n) ⊕ A(n). c) T`ım co. so.’ cu’a S(3) va` A(3). 3. Nho´m coˆ.ng aben Z ca´c soˆ´ nguyeˆn d¯u.o.. c xem nhu. la` moˆ. t moˆd¯un treˆn ch´ınh va`nh Z. a) Ha˜y xa´c d¯i.nh ca´c moˆd¯un con cu’a Z. b) Chu´.ng to’ ra˘`ng khoˆng to`ˆn ta. i hai moˆd¯un con kha´c khoˆng I va` J cu’a Z sao cho Z = I ⊕ J . 4. Cho M la` moˆ. t R-moˆd¯un. M go. i la` khoˆng phaˆn t´ıch d¯u.o.. c neˆ´u khoˆng to`ˆn ta. i hai moˆd¯un con kha´c khoˆng I va` J cu’a M sao cho M la` toˆ’ng tru.. c tieˆ´p cu’a I va` J . Nho´m coˆ.ng aben Z15 ca´c soˆ´ nguyeˆn moˆd¯uloˆ 15 d¯u.o.. c xem nhu. la` moˆd¯un treˆn va`nh ca´c soˆ´ nguyeˆn Z. Ha˜y phaˆn t´ıch Z15 tha`nh toˆ’ng tru.. c tieˆ´p ca´c moˆd¯un con khoˆng phaˆn t´ıch d¯u.o.. c. Su.. phaˆn t´ıch treˆn co´ duy nhaˆ´t khoˆng? 5. Nho´m coˆ.ng aben Q ca´c soˆ´ hu˜.u tı’ d¯u.o.. c xem nhu. la` moˆ. t moˆd¯un treˆn va`nh Z ca´c soˆ´ nguyeˆn. Chu´.ng minh ra`˘ng: a) Hai pha`ˆn tu.’ tuy` y´ cu’a Q la` phu. thuoˆ.c tuyeˆ´n t´ınh treˆn Z. b) Q khoˆng co´ moˆ. t co. so.’ treˆn Z. 6. Chu´.ng minh ra`˘ng tu`. moˆ. t taˆ.p sinh tuy` y´ cu’a Z-moˆd¯un Q, ta ru´t ra moˆ.t pha`ˆn tu.’ baˆ´t ky` th`ı taˆ.p ho.. p ca´c pha`ˆn tu.’ co`n la. i vaˆ˜n la` taˆ.p sinh cu’a Z-moˆd¯un Q. 7. Cho R la` moˆ.t va`nh co´ d¯o.n vi. 1, R d¯u.o.. c xem nhu.R-moˆd¯un tra´i va` mo. i id¯eˆan tra´i cu’a R d¯u.o.. c xem nhu. moˆd¯un con cu’a R- moˆd¯un R. Chu´.ng minh ra˘`ng: a) Neˆ´u I la` moˆ. t id¯eˆan tra´i cu’a R th`ı moˆd¯un thu.o.ng R/I la` moˆ. t R-moˆd¯un cyclic. b) Neˆ´u I la` moˆ. t id¯eˆan tra´i toˆ´i d¯a. i cu’a R th`ı moˆd¯un thu.o.ng R/I la` moˆ. t R-moˆd¯un d¯o.n. 44 8. Cho R la` moˆ. t mie`ˆn nguyeˆn va` M la` moˆ. t R-moˆd¯un. Vo´.i moˆ˜i x ∈M , ky´ hieˆ.u Ann(x) = {r ∈ R | rx = 0} (go. i la` linh hoa´ tu.’ cu’a x). a) Chu´.ng to’ ra˘`ng T (M) = {x ∈M | Ann(x) 6= {0}} la` moˆ. t moˆd¯un con cu’a M , go. i la` moˆd¯un con xoa˘´n cu’a M . b) T´ınh T (M) khi R = Z, M = Z2/L vo´.i L = ((4, 6)). c) M d¯u.o.. c go. i la` khoˆng xoa˘´n neˆ´u T (M) = {0}. Chu´.ng to’ ra˘`ng M/T (M) khoˆng xoa˘´n. d) M d¯u.o.. c go. i la` xoa˘´n neˆ´u T (M) = M . Cho N la` moˆ. t R-moˆd¯un con cu’a M . Chu´.ng to’ ra˘`ng neˆ´u N va` M/N la` xoa˘´n th`ı M la` xoa˘´n. 9. Cho R la` va`nh co´ d¯o.n vi., M la` moˆ.t R-moˆd¯un va` a ∈ R. Xe´t a´nh xa. λa : M −→ M xa´c d¯i.nh bo.’ i λa(x) = ax vo´.i mo. i x ∈ M . Ho’i pha’i cho.n a theˆ´ na`o d¯eˆ’ λa la` moˆ. t tu.. d¯o`ˆng caˆ´u cu’a R-moˆd¯un M? 10. Cho R la` va`nh co´ d¯o.n vi. 1 va` R2 la` moˆd¯un t´ıch treˆn R. Chu´.ng minh ra˘`ng mo. i d¯o`ˆng caˆ´u R-moˆd¯un tu`. R2 va`o R d¯e`ˆu co´ da.ng (x1, x2) 7→ x1a1 + x2a2, vo´.i a1, a2 ∈ R d¯u.o.. c cho.n th´ıch ho.. p. 11. Cho x, y la` ca´c pha`ˆn tu.’ cu’a va`nh R co´ d¯o.n vi. 1 6= 0 thoa’ d¯ie`ˆu kieˆ.n Rx = Ry. Chu´.ng minh ra˘`ng to`ˆn ta. i moˆ. t d¯a˘’ ng caˆ´u R-moˆd¯un pha’i f : xR −→ yR sao cho f(x) = y. 12. Cho hai tru.`o.ng hu˜.u ha.n ca´c soˆ´ nguyeˆn moˆd¯uloˆ Z11 va` Z7. Ta d¯i.nh ngh˜ıa ca´c phe´p toa´n treˆn Z11 ì Z∗7, vo´.i Z∗7 = Z7 \ {0}, nhu. sau: ∀(x, y), (x′, y′) ∈ Z11 ì Z∗7, ∀n ∈ Z, (x, y) + (x′, y′) = (x+ x′, y.y′), n.(x, y) = (nx, yn). a) Chu´.ng to’ ra˘`ng Z11 ì Z∗7 la` moˆ. t moˆd¯un treˆn va`nh ca´c soˆ´ nguyeˆn Z. b) Zn la` nho´m coˆ.ng ca´c soˆ´ nguyeˆn moˆd¯uloˆ n, d¯u.o.. c xem nhu. la` Z-moˆd¯un. Moˆd¯un Z11 ì Z∗7 co´ d¯a˘’ ng caˆ´u vo´.i moˆd¯un Z66 khoˆng? 13. Ky´ hieˆ.u M(2,Z) la` Z-moˆd¯un go`ˆm ca´c ma traˆ.n vuoˆng caˆ´p 2 heˆ. soˆ´ nguyeˆn. Cho f : M(2,Z) −→M(2,Z) la` a´nh xa. xa´c d¯i.nh bo.’ i f(X) = AX −XA, vo´.i A = ( 0 1 2 3 ) . Chu´.ng to’ ra˘`ng f la` moˆ. t d¯o`ˆng caˆ´u Z-moˆd¯un. Ha˜y xa´c d¯i.nh Kerf . 45 14. Ky´ hieˆ.u M(2,Z) la` Z-moˆd¯un go`ˆm ca´c ma traˆ.n vuoˆng caˆ´p 2 heˆ. soˆ´ nguyeˆn. Cho f : M(2,Z) −→M(2,Z) la` a´nh xa. xa´c d¯i.nh bo.’ i f(X) = AX +XA, vo´.i A = ( 1 0 1 1 ) . Chu´.ng to’ ra˘`ng f la` moˆ. t d¯o.n caˆ´u Z-moˆd¯un. 15. Cho R la` moˆ. t va`nh co´ d¯o.n vi., I la` moˆ. t id¯eˆan cu’a R sao cho In = 0 va` M,N la` ca´c R-moˆd¯un pha’i vo´.i f :M −→ N la` moˆ. t d¯o`ˆng caˆ´u R-moˆd¯un. Chu´.ng minh ra˘`ng: a) f ca’m sinh d¯o`ˆng caˆ´u R-moˆd¯un f ′ : M/MI −→ N/NI. b) Neˆ´u f ′ la` moˆ. t toa`n caˆ´u th`ı f cu˜ng la` moˆ. t toa`n caˆ´u. 16. Cho R la` moˆ. t va`nh co´ d¯o.n vi., M la` moˆ. t R-moˆd¯un sao cho M = U ⊕W , trong d¯o´ U,W la` ca´c R-moˆd¯un con cu’a M . Cho ψ : U −→ W la` d¯o`ˆng caˆ´u R-moˆd¯un, ky´ hieˆ.u U1 = {x+ ψ(x) | x ∈ U}. Chu´.ng minh ra˘`ng: a) U1 la` R-moˆd¯un con cu’a M va` U1 ∼= U . b) M = U1 ⊕W . 17. Xem ca´c nho´m cyclic hu˜.u ha.n nhu. nhu˜.ng Z-moˆd¯un. Chu´.ng minh d¯a˘’ ng caˆ´u: HomZ(Zm,Zn) ∼= Z(m,n), trong d¯o´ (m,n) ky´ hieˆ.u u.´o.c chung lo´.n nhaˆ´t cu’a m va` n. 18. Cho R la` moˆ. t va`nh giao hoa´n co´ d¯o.n vi. 1, R d¯u.o.. c xem nhu. la` moˆ.t moˆd¯un treˆn ch´ınh no´ va` M la` moˆ. t R-moˆd¯un. Chu´.ng minh ra˘`ng: HomR(R,M) ∼= M. 19. Cho R la` moˆ. t va`nh co´ d¯o.n vi., M la` moˆ. t R-moˆd¯un, n la` moˆ. t soˆ´ nguyeˆn du.o.ng va` ϕi : M −→ M la` moˆ. t d¯o`ˆng caˆ´u R-moˆd¯un vo´.i mo. i i = 1, . . . , n thoa’ ma˜n: ϕ1 + ã ã ã+ ϕn = idM , ϕi ◦ ϕj = 0 (∀i 6= j). Chu´.ng minh ra˘`ng: a) ϕ2i = ϕi vo´ .i mo. i i = 1, . . . , n. b) M la` toˆ’ng tru.. c tieˆ´p cu’a ca´c moˆd¯un con Mi = Imϕi, i = 1, . . . , n. 20. Cho R la` moˆ. t va`nh co´ d¯o.n vi., 0 −→M ′ f−→M g−→M ′′ −→ 0 la` da˜y kho´.p ca´c d¯o`ˆng caˆ´u R-moˆd¯un (ngh˜ıa la` f d¯o.n caˆ´u, g toa`n caˆ´u va` Imf = Kerg) va` f kha’ nghi.ch tra´i, tu´.c la` co´ d¯o`ˆng caˆ´u R-moˆd¯un ψ : M −→ M ′ sao cho ψ ◦ f = idM ′ . Chu´.ng minh ra˘`ng: 46 a) M = Imf ⊕Kerψ. b) g kha’ nghi.ch pha’i, tu´.c la` co´ d¯o`ˆng caˆ´u R-moˆd¯un ϕ : M ′′ −→ M sao cho g ◦ ϕ = idM ′′ . 21. Cho R la` moˆ. t va`nh co´ d¯o.n vi., 0 −→M ′ f−→M g−→M ′′ −→ 0 la` da˜y kho´.p ca´c d¯o`ˆng caˆ´u R-moˆd¯un (ngh˜ıa la` f d¯o.n caˆ´u, g toa`n caˆ´u va` Imf = Kerg) va` g kha’ nghi.ch pha’i, tu´.c la` co´ d¯o`ˆng caˆ´u R-moˆd¯un ϕ : M ′′ −→M sao cho g ◦ ϕ = idM ′′ . Chu´.ng minh ra˘`ng: a) M = Kerg ⊕ Imϕ. b) f kha’ nghi.ch tra´i, tu´.c la` co´ d¯o`ˆng caˆ´u R-moˆd¯un ψ : M −→ M ′ sao cho ψ ◦ f = idM ′ . 22. Cho A, B, C, D la` ca´c R-moˆd¯un va` ca´c d¯o`ˆng caˆ´u R-moˆd¯un α : A −→ B, β : B −→ D γ : A −→ C, δ : C −→ D sao cho β ◦ α = δ ◦ γ. Chu´.ng minh ra˘`ng neˆ´u γ la` toa`n caˆ´u va` β la` d¯o.n caˆ´u, ta co´: a) Im(α) = β−1(Im(δ)). b) Ker(δ) = γ(Ker(α)). 23. a) Cho ϕ : A −→ A la` moˆ. t d¯o`ˆng caˆ´u R-moˆd¯un thoa’ ma˜n ϕ ◦ϕ = ϕ. Chu´.ng minh ra˘`ng A = Imϕ ⊕Kerϕ. b) Cho ϕ : A −→ B va` ψ : B −→ C la` hai d¯o`ˆng caˆ´u R-moˆd¯un sao cho ψ ◦ϕ la` moˆ. t d¯a˘’ ng caˆ´u. Chu´.ng minh ra˘`ng B = Imϕ⊕Kerψ. 24. Cho bieˆ’u d¯o`ˆ ca´c d¯o`ˆng caˆ´u R-moˆd¯un sau: A f−−−−→ B g−−−−→ C −−−−→ 0yh D trong d¯o´ do`ng la` kho´.p va` h ◦ f = 0. Chu´.ng minh ra`˘ng to`ˆn ta. i moˆ. t d¯o`ˆng caˆ´u R-moˆd¯un duy nhaˆ´t k : C −→ D sao cho k ◦ g = h. 25. Cho bieˆ’u d¯o`ˆ ca´c d¯o`ˆng caˆ´u R-moˆd¯un sau: Dyh 0 −−−−→ A f−−−−→ B g−−−−→ C trong d¯o´ do`ng la` kho´.p va` g ◦ h = 0. Chu´.ng minh ra`˘ng to`ˆn ta. i moˆ. t d¯o`ˆng caˆ´u R-moˆd¯un duy nhaˆ´t k : D −→ A sao cho f ◦ k = h. 47 TRA’ LO` . I VA` HU . O´ . NG DA˜ˆN GIA’ I BA`I TAˆ. P CHU . O . NG III – MOˆD- UN 1. ∀(x, y), (u, v), (t, w) ∈M, ∀α, β ∈ R, (x, y) + (u, v) = (x+ u, yv) = (u+ x, vy) = (u, v) + (x, y) ((x, y) + (u, v)) + (t, w) = (x+ u, yv) + (t, w) = (x+ u+ t, yvw) = (x, y) + (u+ t, vw) = (x, y) + ((u, v) + (t, w)) (x, y) + (0, 1) = (x+ 0, y1) = (x, y) (x, y) + (−x, y−1) = (x− x, yy−1) = (0, 1) α((x, y) + (u, v)) = α(x+ u, yv) = (α(x+ u), (yv)α) = (αx+ αu, yαvα) = (αx, yα) + (αu, vα) = α(x, y) + α(u, v) (α+ β)(x, y) = ((α+ β)x, yα+β) = (αx+ βx, yαyβ) = (αx, yα) + (βx, yβ) = α(x, y) + β(x, y) α(β(x, y)) = α(βx, yβ) = (αβx, (yβ)α) = αβ(x, y) 1(x, y) = (1.x, y1) = (x, y) 2. a) Ta co´ ma traˆ.n 0 thuoˆ.c S(n) va` A(n) neˆn S(n) 6= ∅ va` A(n) 6= ∅. ∀A,B ∈ S(n) (t.u.. A,B ∈ A(n)), ∀α, β ∈ R, (αA+βB)t = (αA)t+(βB)t = αAt+βBt = αA+βB hay αA+βB ∈ S(n) (t.u..(αA+βB)t = (αA)t+(βB)t = αAt+βBt = −αA−βB hay αA+βB ∈ A(n)). Vaˆ.y S(n) va` A(n) la` ca´c moˆd¯un con cu’a Mn(R). b) ∀A ∈Mn(R), d¯a˘. t B = A+ At 2 va` C = A− At 2 . Ta co´ Bt = At + (At)t 2 = At + A 2 = B hay B ∈ S(n) Ct = At − (At)t 2 = At − A 2 = −C hay C ∈ A(n) A = B + C. Do d¯o´ Mn(R) = S(n) + A(n). Ma˘. t kha´c, 48 A ∈ S(n) ∩ A(n) ⇒ { At = A At = −A ⇒ A = −A⇒ A = 0 ⇒ S(n) ∩ A(n) = {0}. Vaˆ.y Mn(R) = S(n)⊕ A(n). c) Co. so.’ cu’a S(3) va` A(3) la`ˆn lu.o.. t la`:( 1 0 00 0 0 0 0 0  ,  0 0 00 1 0 0 0 0  ,  0 0 00 0 0 0 0 1  ,  0 1 01 0 0 0 0 0  ,  0 0 10 0 0 1 0 0  ,  0 0 00 0 1 0 1 0 ), ( 0 1 0−1 0 0 0 0 0  ,  0 0 10 0 0 −1 0 0  ,  0 0 00 0 1 0 −1 0 ). 3. a) Cho I la` moˆ. t moˆd¯un con cu’a Z. Neˆ´u I 6= {0} th`ı I chu´.a ı´t nhaˆ´t moˆ. t soˆ´ nguyeˆn du.o.ng. Go. i n la` soˆ´ nguyeˆn du.o.ng nho’ nhaˆ´t sao cho n ∈ I. Khi d¯o´ ∀m ∈ I, m = nq + r, vo´.i 0 ≤ r < n. Do r = m − nq ∈ I va` t´ınh nho’ nhaˆ´t cu’a n, ta co´ r = 0 hay m = nq ∈ nZ = {nk | k ∈ Z}. Vo´.i n ∈ N, nZ la` moˆ. t moˆd¯un con cu’a Z. Thaˆ. t vaˆ. y, ro˜ ra`ng nZ 6= ∅. ∀x, y ∈ nZ, ∀a, b ∈ Z, ∃k, l ∈ Z, x = nk, y = nl, ta co´ ax+by = ank+bnl = n(ak+bl) ∈ nZ. b) Gia’ su.’ Z = I⊕J vo´.i I va` J la` hai moˆd¯un con kha´c khoˆng cu’a Z. Khi d¯o´ to`ˆn ta. i hai soˆ´ nguyeˆn du.o.ng n va` m sao cho I = nZ va` J = mZ. Ta co´ nm 6= 0 va` nm ∈ nZ∩mZ = I ∩ J = {0}. D- ie`ˆu maˆu thuaˆ’n na`y cho bieˆ´t Z khoˆng la` toˆ’ng tru.. c tieˆ´p cu’a hai moˆd¯un con kha´c khoˆng I va` J . 4. Ca´c moˆd¯un con cu’a Z-moˆd¯un Z15 la` {0}, 3Z15 = {0, 3, 6, 9, 12}, 5Z15 = {0, 5, 10}, O.’ d¯aˆy, 3Z15, 5Z15 khoˆng co´ moˆd¯un con kha´c khoˆng na`o neˆn chu´ng la` ca´c moˆd¯un con khoˆng phaˆn t´ıch d¯u.o.. c. Ngoa`i ra, ta co´ 1 = 6 + 10. Do d¯o´ Z15 = 3Z15 ⊕ 5Z15 va` d¯aˆy la` su.. phaˆn t´ıch duy nhaˆ´t tha`nh toˆ’ng tru.. c tieˆ´p ca´c moˆd¯un con khoˆng phaˆn t´ıch d¯u.o.. c. 5. a) Neˆ´u laˆ´y hai pha`ˆn tu.’ , trong d¯o´ co´ pha`ˆn tu.’ 0 th`ı hieˆ’n nhieˆn chu´ng phu. thuoˆ.c tuyeˆ´n t´ınh. 49 Neˆ´u laˆ´y hai pha`ˆn tu.’ kha´c 0: x = a b , y = c d , (a, b, c, d ∈ Z \ {0}). Ta co´ bc a b = da c d ⇔ αx+ βy = 0, vo´.i α = bc, β = −da ∈ Z \ {0}. Vaˆ.y x, y phu. thuoˆ.c tuyeˆ´n t´ınh. b) Theo treˆn, muoˆ´n co´ moˆ. t Z-co. so.’ cho Q th`ı co. so.’ d¯o´ chı’ co´ theˆ’ co´ 1 pha`ˆn tu.’ . Gia’ su.’ Q =< a b > (a, b ∈ Z). Nhu.ng d¯ie`ˆu na`y khoˆng theˆ’ d¯u.o.. c v`ı neˆ´u n = a b ∈ Z th`ı < a b >== nZ 6= Q, co`n neˆ´u a b /∈ Z th`ı < a b >= {na b | n ∈ Z} 6= Q. 6. Cho X la` moˆ. t taˆ.p sinh cu’a Z-moˆd¯un Q. Laˆ´y x0 ∈ X tuy` y´ va` ru´t no´ ra kho’i X. Khi d¯o´ x0 2 co´ theˆ’ bieˆ’u die˜ˆn tha`nh moˆ.t toˆ’ng hu˜.u ha.n la`: x0 2 = z0x0 + ∑ xi 6=x0 zixi, xi ∈ X, zi ∈ Z. Tu`. d¯o´ x0 = 2z0x0 + ∑ xi 6=x0 2zixi va` nx0 = ∑ xi 6=x0 2zixi, trong d¯o´ n = 1 − 2z0 ∈ Z, n 6= 0. Tieˆ´p tu. c, x0 n co´ bieˆ’u die˜ˆn tha`nh toˆ’ng hu˜.u ha.n: x0 n = z′0x0 + ∑ xi 6=x0 z′ixi, xi ∈ X, z′i ∈ Z. Khi d¯o´ x0 = nz′0x0 + ∑ xi 6=x0 nz′ixi = ∑ xi 6=x0 2ziz′0xi + ∑ xi 6=x0 nz′ixi = ∑ xi 6=x0 z′′i xi, xi ∈ X, z′′i = 2ziz ′ 0 +nz ′ i ∈ Z. D- ie`ˆu na`y cho bieˆ´t x0 d¯u.o.. c bieˆ’u die˜ˆn qua taˆ.p X \{x0}. Do X la` heˆ. sinh cu’a Q neˆn X \ {x0} cu˜ng la` heˆ. sinh cu’a Q. 7. a) Moˆ˜i pha`ˆn tu.’ cu’a moˆd¯un thu.o.ng R/I co´ da.ng x + I, vo´.i x ∈ R va` x+ I = x.1 + I = x(1 + I). Do d¯o´ R/I la` moˆ. t R-moˆd¯un cyclic sinh bo.’ i 1 + I. b) Moˆ˜i moˆd¯un con cu’a moˆd¯un thu.o.ng R/I co´ da.ng J/I, vo´.i J la` id¯eˆan tra´i cu’a R va` chu´.a I. Do I la` cu.. c d¯a. i, ta co´ J = R hoa˘.c J = I, tu´.c la` J/I hoa˘.c la` moˆd¯un R/I hoa˘.c la` moˆd¯un khoˆng. Vaˆ.y R/I la` moˆ. t R-moˆd¯un d¯o.n. 8. a) Ro˜ ra`ng 0 ∈ T (M) hay T (M) 6= ∅. ∀x, y ∈ T (M), ∀α, β ∈ R, ∃r, s ∈ R, r 6= 0, s 6= 0 sao cho rx = 0 va` sy = 0. Khi d¯o´ do R la` moˆ. t mie`ˆn nguyeˆn neˆn rs 6= 0 va` rs(αx + βy) = sα(rx) + rβ(sy) = 0 + 0 = 0 hay αx+ βy ∈ T (M). Vaˆ.y T (M) la` moˆ. t moˆd¯un con cu’a M . 50 b) T (M) = {(k, l) + L ∈ Z2/L | ∃n ∈ Z \ {0}, n(k, l) ∈ ((4, 6))} = {(k, l) + L ∈ Z2/L | ∃n ∈ Z \ {0}, ∃m ∈ Z, n(k, l) = m(4, 6)} = {(k, l) + L ∈ Z2/L | ∃n ∈ Z \ {0}, nk 2 = nl 3 ∈ Z} = {(k, l) + L ∈ Z2/L | 3k = 2l} c) x+ T (M) ∈ T (M/T (M)) ⇒ ∃r ∈ R \ {0}, r(x+ T (M)) = T (M) ⇒ rx ∈ T (M) ⇒ ∃s ∈ R \ {0}, s(rx) = 0 ⇒ ∃rs ∈ R \ {0}, (rs)x = 0 ⇒ x ∈ T (M) ⇒ x+ T (M) = T (M) ⇒ T (M/T (M)) = {T (M)} Vaˆ.y M/T (M) khoˆng xoa˘´n. d) N va` M/N xoa˘´n ⇒ ∀x ∈M, ∃r ∈ R \ {0}, r(x +N) = N ⇒ ∀x ∈M, ∃r ∈ R \ {0}, rx ∈ N ⇒ ∀x ∈M, ∃r, s ∈ R \ {0}, s(rx) = 0 ⇒ ∀x ∈M, ∃rs ∈ R \ {0}, (rs)x = 0 ⇒M xoa˘´n 9. ∀x, y ∈ M, λa(x + y) = a(x + y) = ax + ay = λa(x) + λa(y). Do d¯o´ λa la` moˆ. t d¯o`ˆng caˆ´u R-moˆd¯un khi va` chı’ khi ∀r ∈ R, ∀x ∈ M, λa(rx) = rλa(x) hay a(rx) = r(ax) hay (ar − ra)x = 0. Nhu. vaˆ.y λa ∈ HomR(M,M) ⇔ a ∈ {α ∈ R | (αr − rα)x = 0, ∀r ∈ R, ∀x ∈M}. 10. Cho f : R2 −→ R xa´c d¯i.nh bo.’ i f(x1, x2) = x1a1 + x2a2, vo´.i a1, a2 ∈ R na`o d¯o´. Khi d¯o´ ∀(x1, x2), (y1, y2) ∈ R2, ∀r, s ∈ R, f(r(x1, x2) + s(y1, y2)) = f(rx1 + sy1, rx2 + sy2) = (rx1 + sy1)a1 + (rx2 + sy2)a2 = r(x1a1 + x2a2) + s(y1a1 + y2a2) = rf(x1, x2) + sf(y1, y2). 51 Do d¯o´ f la` moˆ. t d¯o`ˆng caˆ´u R-moˆd¯un. D- a’o la. i, cho f : R2 −→ R la` moˆ. t d¯o`ˆng caˆ´u R-moˆd¯un. Khi d¯o´ ∀(x1, x2) ∈ R2, d¯a˘. t a1 = f(1, 0), a2 = f(0, 1) ∈ R, ta co´: f(x1, x2) = f((x1, 0) + (0, x2)) = f(x1(1, 0) + x2(0, 1)) = x1f(1, 0) + x2f(0, 1) = x1a1 + x2a2. 11. Go. i f : xR −→ yR xa´c d¯i.nh bo.’ i f(xr) = yr vo´.i r ∈ R. Neˆ´u xr = xr′ vo´.i r, r′ ∈ R th`ı x(r − r′) = 0, do y ∈ Ry = Rx neˆn y = r′′x vo´.i r′′ ∈ R, v`ı vaˆ. y y(r − r′) = r′′x(r − r′) = r′′0 = 0 hay yr = yr′. Do d¯o´ f la` moˆ. t a´nh xa. . ∀r, r′, s, s′ ∈ R, f((xr)s+(xr′)s′) = f(x(rs+ r′s′)) = y(rs+ r′s′) = (yr)s+ (yr′)s′ = f(xr)s + f(xr′)s′. Do d¯o´ f la` moˆ. t d¯o`ˆng caˆ´u R-moˆd¯un pha’i. Ro˜ ra`ng f la` moˆ. t toa`n a´nh. Cho xr ∈ Kerf hay yr = 0. Do x ∈ Rx = Ry neˆn x = r′y vo´.i r′ ∈ R neˆn xr = r′yr = r′0 = 0. Vaˆ.y Kerf = {0} hay f la` moˆ. t d¯o.n caˆ´u. Vaˆ.y f la` moˆ. t d¯a˘’ ng caˆ´u R-moˆd¯un pha’i. 12. a) Phe´p coˆ.ng treˆn Z11 ì Z∗7 co´ t´ınh giao hoa´n, keˆ´t ho.. p, co´ pha`ˆn tu.’ khoˆng la` (0, 1) va` moˆ˜i pha`ˆn tu.’ (x, y) ∈ Z11ìZ∗7 co´ pha`ˆn tu.’ d¯oˆ´i la` (−x, y−1). Ngoa`i ra, ∀(x, y), (x′, y′) ∈ Z11 ì Z∗7, ∀n,m ∈ Z,, n((x, y)+(x′, y′)) = n(x+x′, yy′) = (n(x+x′), (yy′)n) = (nx+nx′, yny ′n) = (nx, yn) + (nx′, y ′n) = n(x, y) + n(x′, y′), (n+m)(x, y) = ((n+m)x, yn+m) = (nx+mx, ynym) = (nx, yn)+(mx, ym) = n(x, y) +m(x, y), n(m(x, y)) = n(mx, ym) = (nmx, ymn) = nm(x, y), 1(x, y) = (1x, y1) = (x, y). Vaˆ.y Z11 ì Z∗7 la` moˆ. t Z-moˆd¯un. b) Z11 la` nho´m coˆ.ng cyclic caˆ´p 11, sinh bo.’ i 0 va` Z∗7 la` nho´m nhaˆn cyclic caˆ´p 6 sinh bo.’ i 3. Vı` (11, 6) = 1 neˆn Z11 ì Z∗7 la` nho´m cyclic caˆ´p 11.6=66. Vı` vaˆ. y, Z11 ì Z∗7 d¯a˘’ ng caˆ´u Z- moˆd¯un vo´.i Z66. 13. ∀X,Y ∈M(2,Z), ∀a, b ∈ Z, f(aX + bY ) = A(aX + bY )− (aX + bY )A = A(aX) +A(bY )− (aX)A− (bY )A = a(AX) + b(AY )− a(XA)− b(Y A) = a(AX −XA) + b(AY − Y A) = af(X) + bf(Y ). Do d¯o´ f la` moˆ. t d¯o`ˆng caˆ´u Z-moˆd¯un. Kerf = {( a b c d ) ∈M(2,Z) | f (( a b c d )) = ( 0 0 0 0 )} 52 Kerf = {( a b c d ) ∈M(2,Z) | ( 0 1 2 3 )( a b c d ) − ( a b c d )( 0 1 2 3 ) = ( 0 0 0 0 )} = {( a b c d ) ∈M(2,Z) | ( c− 2b d− a− 3b 2a+ 3c− 2d 2b− c ) = ( 0 0 0 0 )} = {( a b c d ) ∈M(2,Z) | c = 2b, d = a+ 3b } = {( a b 2b a+ 3b ) | a, b ∈ Z } = 〈(1 0 0 1 ) , ( 0 1 2 3 )〉 . 14. ∀X,Y ∈M(2,Z), ∀a, b ∈ Z, f(aX + bY ) = A(aX + bY ) + (aX + bY )A = A(aX) +A(bY ) + (aX)A+ (bY )A = a(AX) + b(AY ) + a(XA) + b(Y A) = a(AX +XA) + b(AY + Y A) = af(X) + bf(Y ). Do d¯o´ f la` moˆ. t d¯o`ˆng caˆ´u Z-moˆd¯un. Kerf = {( a b c d ) | ( 1 0 1 1 )( a b c d ) + ( a b c d )( 1 0 1 1 ) = ( 0 0 0 0 )} = {( a b c d ) | ( 2a+ b 2b a+ 2c+ d b+ 2d ) = ( 0 0 0 0 )} = {(0 0 0 0 )} . Vaˆ.y f la` moˆ. t d¯o.n caˆ´u Z-moˆd¯un. 15. a) Vo´.i x ∈ MI, ta co´ x = n∑ i=1 miai, trong d¯o´ mi ∈ M, ai ∈ I. Khi d¯o´ f(x) = n∑ i=1 f(mi)ai, neˆn f(x) ∈ NI. Tu`. d¯o´ vo´.i x1, x2 ∈M, x1+MI = x2+MI, ta co´ x1 − x2 ∈ MI, do d¯o´ f(x1 − x2) ∈ NI hay f(x1) + NI = f(x2) +NI. Vı` vaˆ. y, ta co´ a´nh xa. f ′ : M/MI −→ N/NI xa´c d¯i.nh bo.’ i f(x+MI) = f(x) + NI. Kieˆ’m tra d¯o`ˆng caˆ´u R-moˆd¯un la` de˜ˆ da`ng. b) Tu`. laˆ.p luaˆ.n treˆn, f(MI2) ⊂ f(NI2) va` ca’m sinh d¯o`ˆng caˆ´u R-moˆd¯un f ′′ : M/MI2 −→ N/NI2. Ta chu´.ng minh ra`˘ng neˆ´u f ′ la` moˆ. t toa`n caˆ´u th`ı f ′′ cu˜ng vaˆ.y. 53 Cho y ∈ N , v`ı f ′ la` moˆ. t toa`n caˆ´u, to`ˆn ta. i x ∈M sao cho y+NI = f(x)+NI. Vaˆ. y to`ˆn ta. i a1, . . . , an ∈ I va` y1, . . . , yn ∈ N sao cho y = f(x) + n∑ i=1 yiai. La`m tu.o.ng tu.. vo´.i moˆ˜i yi, to`ˆn ta.i x1, . . . , xn ∈ M ma` yi = f(xi) + zi, zi ∈ NI. Tu`. d¯o´ suy ra y = f(x+ n∑ i=1 xiai) + z, vo´.i z = n∑ i=1 ziai ∈ NI2. Do d¯o´ f ′′ la` mứt toa`n caˆ´u. Quy na.p ta co´ f = f (n) : M = M/MIn −→ N/NIn = N la` moˆ. t toa`n caˆ´u. 16. a) 0 = 0 + ψ(0) ∈ U1 neˆn U1 6= ∅. ∀a, b ∈ U1, ∀α, β ∈ R, a = x+ ψ(x), b = y + ψ(y) vo´.i x, y ∈ U , ta co´: αa+ βb = α(x+ ψ(x)) + β(y + ψ(y)) = (αx+ βy) + ψ(αx+ βy). Vı` x, y ∈ U neˆn αx+ βy ∈ U , do d¯o´ αa+ βb ∈ U1. Vaˆ.y U1 la` moˆ. t moˆd¯un con cu’a M . Xe´t a´nh xa. ϕ : U −→ U1 cho bo.’ i ϕ(x) = x+ψ(x). Khi d¯o´ ∀x, y ∈ U, ∀α, β ∈ R, ta co´: ϕ(αx+βy) = αx+βy+ψ(αx+βy) = α(x+ψ(x))+β(y+ψ(y)) = αϕ(x)+βϕ(y). Vaˆ.y ϕ la` moˆ. t d¯o`ˆng caˆ´u R-moˆd¯un. Ro˜ ra`ng ϕ la` moˆ. t toa`n caˆ´u. Ngoa`i ra, x ∈ Kerϕ⇒ ϕ(x) = 0 ⇒ x = −ψ(x) ⇒ x ∈ U ∩W = {0} ⇒ x = 0. Do d¯o´ Kerϕ = {0} hay ϕ la` moˆ. t d¯o.n caˆ´u. Vaˆ. y ϕ la` moˆ. t d¯a˘’ ng caˆ´u. b) ∀z ∈ M, z = x+ y vo´.i x ∈ U, y ∈ W, z = (x+ ψ(x)) + (y − ψ(x)). Vı` x+ ψ(x) ∈ U1, y − ψ(x) ∈W neˆn z ∈ U1 +W . Do d¯o´ M = U1 +W . z ∈ U1 ∩W ⇒ z = x+ ψ(x) ∈W, x ∈ U ⇒ x = z − ψ(x) ∈ U ∩W ⇒ x = 0. Do d¯o´ z = 0 hay U1 ∩W = {0}. Vaˆ. y M = U1 ⊕W . 17. Moˆ˜i d¯o`ˆng caˆ´u f : Zm −→ Zn xa´c d¯i.nh gia´ tri. a = f(1) ∈ Zn va` do ma = mf(1) = f(m1) = f(m) = f(0) = 0 neˆn ta co´ caˆ´p cu’a a trong Zn la` moˆ. t u.´o.c cu’a m (do d¯o´ la` moˆ. t u.´o.c chung cu’a m va` n). D- a’o la. i, pha`ˆn tu.’ a ∈ Zn co´ caˆ´p la` moˆ. t u.´o.c cu’a m th`ı phe´p tu.o.ng u´.ng f : Zm −→ Zn : k 7→ ka la` moˆ. t a´nh xa. va` khi d¯o´ f la` moˆ. t d¯o`ˆng caˆ´u Z-moˆd¯un. D- a˘. t d = (m,n), d′ = n d . Khi d¯o´ HomZ(Zm,Zn) = {fi | 0 ≤ i ≤ d− 1}. 54 O.’ d¯aˆy, fi : Zm −→ Zn cho bo.’ i fi(1) = id′ va` fi = if1. Vaˆ. y, HomZ(Zm,Zn) ∼= d′Zn ∼= Zd. 18. Xe´t a´nh xa. ϕ : HomR(R,M) −→M cho bo.’ i ϕ(f) = f(1). Khi d¯o´: ∀f, g ∈ HomR(R,M), ∀a, b ∈ R, ϕ(af + bg) = (af + bg)(1) = af(1) + bg(1) = aϕ(f) + bϕ(g). Do d¯o´ ϕ la` moˆ. t d¯o`ˆng caˆ´u R-moˆd¯un. ∀f ∈ HomR(R,M), ϕ(f) = f(1) = 0 ke´o theo f(a) = f(a.1) = af(1) = a0 = 0, ∀a ∈ R. Do d¯o´ Kerϕ = {0} hay ϕ la` moˆ. t d¯o.n caˆ´u. ∀x ∈ M , xe´t a´nh xa. fx : R −→ M cho bo.’ i fx(a) = ax. Khi d¯o´ ∀a, b, α, β ∈ R, fx(αa + βb) = (αa + βb)x = α(ax) + β(bx) = αfx(a) + βfx(b) hay fx ∈ HomR(R,M) va` ϕ(fx) = fx(1) = 1.x = x. Do d¯o´ ϕ la` moˆ. t toa`n caˆ´u. Vaˆ.y ϕ la` moˆ. t d¯a˘’ ng caˆ´u R-moˆd¯un. 19. a) Vo´.i mo. i i = 1, . . . , n, ta co´ ϕi = ϕi ◦ idM = ϕi ◦ (ϕ1 + ã ã ã+ ϕn) = ϕi ◦ ϕi = ϕ2i . b) ∀x ∈ M, x = idM(x) = (ϕ1 + ã ã ã + ϕn)(x) = ϕ1(x) + ã ã ã + ϕn(x), vo´.i ϕi(x) ∈Mi. Do d¯o´ M = M1 + ã ã ã+Mn. Vo´.i moˆ˜i i = 1, . . . , n, ∀x ∈ Mi ∩ ∑ j 6=i Mj , ta co´ x = xi = ∑ j 6=i xj, xk ∈ Mk = ϕk(M) (1 ≤ k ≤ n), neˆn ∃yk ∈M sao cho xk = ϕk(yk) va` ϕk(xk) = ϕ2k(yk) = ϕk(yk) = xk, x = xi = ϕi(xi) = ϕi( ∑ j 6=i ϕj(xj)) = ∑ j 6=i ϕi ◦ ϕj(xj) = 0. Do d¯o´ Mi ∩ ∑ j 6=i Mj = {0}. Vaˆ. y M = M1 ⊕ ã ã ã ⊕Mn. 20. a) ∀x ∈ M , d¯a˘. t y = f(ψ(x)), z = x − y, ta co´ ψ(z) = ψ(x − f(ψ(x))) = ψ(x) − ψ(f(ψ(x))) = ψ(x) − ψ(x) = 0 hay z ∈ Kerψ. Do d¯o´ ∀x ∈ M, ∃y ∈ Imf, ∃z ∈ Kerψ, x = y + z hay M = Imf + Kerψ. x ∈ Imf ∩Kerψ ⇒ x = f(u), u ∈M ′ va` ψ(x) = 0 ⇒ u = ψ(f(u)) = ψ(x) = 0 ⇒ x = 0. Do d¯o´ Imf ∩Kerψ = {0}. Vaˆ. y M = Imf ⊕Kerψ. 55 b) Xe´t d¯o`ˆng caˆ´u g′ = g ∣∣ Kerψ : Kerψ −→M ′′. Ta co´: g′(Kerψ) = g(Imf ⊕Kerψ) = g(M) = M ′′ Kerg′ = Kerg ∩Kerψ = Imf ∩Kerψ = {0}. Do d¯o´ g′ la` moˆ. t d¯a˘’ ng caˆ´u R-moˆd¯un. D- a˘. t ϕ = (g′)−1, ta co´ ϕ la` moˆ. t d¯o`ˆng caˆ´u R-moˆd¯un va` kieˆ’m tra de˜ˆ da`ng g ◦ ϕ = idM ′′ . 21. a) ∀x ∈ M , d¯a˘. t y = ϕ(g(x)), z = x − y, ta co´ g(z) = g(x − ϕ(g(x))) = g(x) − g(ϕ(g(x))) = g(x) − g(x) = 0 hay z ∈ Kerg. Do d¯o´ ∀x ∈ M, ∃z ∈ Kerg, ∃y ∈ Imϕ, x = z + y hay M = Kerg + Imϕ. x ∈ Kerg ∩ Imϕ⇒ x = ϕ(u), u ∈M ′′ va` g(x) = 0 ⇒ u = g(ϕ(u)) = g(x) = 0 ⇒ x = 0. Do d¯o´ Kerg ∩ Imϕ = {0}. Vaˆ.y M = Kerg ⊕ Imϕ. b) Do f la` d¯o.n caˆ´u va` Imf = Kerg, neˆn f : M ′ −→ Kerg la` moˆ.t d¯a˘’ ng caˆ´u. Xe´t a´nh xa. ψ : M −→ M ′ xa´c d¯i.nh bo.’ i ψ ∣∣ Kerg = f−1, ψ ∣∣ Imϕ = 0. Khi d¯o´ ψ la` moˆ. t d¯o`ˆng caˆ´u R-moˆd¯un va` kieˆ’m tra de˜ˆ da`ng ψ ◦ f = idM ′ . 22. a) b ∈ Imα⇒ ∃a ∈ A, b = α(a) ⇒ β(b) = β(α(a)) = δ(γ(a)) ∈ Imδ ⇒ b ∈ β−1(Imδ). Do d¯o´ Imα ⊂ β−1(Imδ). b ∈ β−1(Imδ) ⇒ β(b) ∈ Imδ ⇒ ∃c ∈ C, β(b) = δ(c) ⇒ ∃a ∈ A, γ(a) = c va` β(b) = δ(γ(a)) = β(α(a)) ⇒ b = α(a) ∈ Imα. Do d¯o´ β−1(Imδ) ⊂ Imα. Vaˆ.y Imα = β−1(Imδ). b) c ∈ Kerδ ⇒ ∃a ∈ A, γ(a) = c va` δ(c) = 0 ⇒ β(α(a)) = δ(γ(a)) = δ(c) = 0 ⇒ α(a) = 0 ⇒ a ∈ Kerα va` c = γ(a) ∈ γ(Kerα). Do d¯o´ Kerδ ⊂ γ(Kerα). c ∈ γ(Kerα) ⇒ ∃a ∈ Kerα, γ(a) = c ⇒ α(a) = 0 va` δ(c) = δ(γ(a)) = β(α(a)) = β(0) = 0 ⇒ c ∈ Kerδ. Do d¯o´ γ(Kerα) ⊂ Kerδ. Vaˆ.y Kerδ = γ(Kerα). 23. a) ∀x ∈ A, d¯a˘. t y = ϕ(x) va` z = x − y. Khi d¯o´ x = y + z vo´.i y ∈ Imϕ va` z ∈ Kerϕ do ϕ(z) = ϕ(x) − ϕ(y) = ϕ(x) − ϕ(ϕ(x)) = ϕ(x) − ϕ(x) = 0. Do d¯o´ A = Imϕ+Kerϕ. x ∈ Imϕ ∩ Kerϕ ⇒ ∃u ∈ A, ϕ(u) = x va` ϕ(x) = 0 ⇒ x = ϕ(u) = ϕ(ϕ(u)) = ϕ(x) = 0. Do d¯o´ Imϕ ∩Kerϕ = {0}. Vaˆ.y A = Imϕ ⊕Kerϕ. b) ∀x ∈ B, ψ(x) ∈ C, neˆn ∃u ∈ A sao cho ψ ◦ ϕ(u) = ψ(x). Khi d¯o´ y = ϕ(u) ∈ Imϕ va` vo´.i z = x− y ta co´ ψ(z) = ψ(x) − ψ(y) = 0 hay z ∈ Kerψ. Vaˆ. y ∀x ∈ B, x = y + z vo´.i y ∈ Imϕ va` z ∈ Kerψ hay B = Imϕ +Kerψ. x ∈ Imϕ ∩Kerψ ⇒ ∃u ∈ A, x = ϕ(u) va` ψ ◦ ϕ(u) = ψ(x) = 0 ⇒ u = 0 va` x = ϕ(0) = 0. Vaˆ.y Imϕ ∩Kerψ = {0}. Do d¯o´ B = Imϕ ⊕Kerψ. 56 24. Do g la` toa`n caˆ´u neˆn ∀c ∈ C, ∃b ∈ B, g(b) = c. Khi d¯o´ ta co´ a´nh xa. k : C −→ D xa´c d¯i.nh bo.’ i k(c) = h(b); thaˆ. t vaˆ. y, neˆ´u ∃b, b′ ∈ B sao cho g(b) = g(b′) = c th`ı b− b′ ∈ Kerg = Imf do d¯o´ ∃a ∈ A sao cho b− b′ = f(a) va` h(b)− h(b′) = h(b− b′) = h(f(a)) = 0 hay h(b) = h(b′). ∀c, c′ ∈ C, ∀α,α′ ∈ R, ∃b, b′ ∈ B sao cho g(b) = c, g(b′) = c′, neˆn g(αb + α′b′) = αc + α′c′. Khi d¯o´ k(αc + α′c′) = h(αb + α′b′) = αh(b) + α′h(b′) = αk(c) + αk(c′), do d¯o´ k la` moˆ. t d¯o`ˆng caˆ´u R-moˆd¯un. Tu`. d¯i.nh ngh˜ıa cu’a k, ta co´ k ◦ g = h va` neˆ´u to`ˆn ta. i k′ : C −→ D sao cho k′ ◦ g = h th`ı k = k′. 25. ∀d ∈ D, g(h(d)) = 0, neˆn h(d) ∈ Kerg = Imf , do d¯o´ ∃a ∈ A sao cho f(a) = h(d). Xe´t phe´p tu.o.ng u´.ng k : D −→ A cho bo.’ i k(d) = a vo´.i f(a) = h(d). k la` moˆ.t a´nh xa. v`ı neˆ´u ∃a, a′ sao cho f(a) = f(a′) = h(d) th`ı a = a′ (do f la` d¯o.n caˆ´u). ∀d, d′ ∈ D, ∀λ, λ′ ∈ R, ∃a, a′ ∈ A sao cho k(d) = a, k(d′) = a′ vo´.i f(a) = h(d), f(a′) = h(d′), neˆn f(λa+λ′a′) = h(λd+λ′d′). Khi d¯o´ k(λd+λ′d′) = λa+ λ′a′ = λk(d) + λ′k(d′), do d¯o´ k la` moˆ. t d¯o`ˆng caˆ´u R-moˆd¯un. Tu`. d¯i.nh ngh˜ıa cu’a k, ta co´ f ◦ k = h va` neˆ´u to`ˆn ta. i k′ : D −→ A sao cho f ◦ k′ = h th`ı k = k′. 57