b) T`ım c´ac phˆ` n tu.a ’ kha’ nghi.ch cu’a S.
32. Ch´u.ng minh r˘a`ng id¯ˆean ch´ınh (x2−x+ 1) l`a id¯ˆean cu.. c d¯a.i cu’a v`anh R[x]v´o.iR l`a tru.`o.ng c´ac sˆo´ thu.. c. T`u. d¯´o suy ra v`anh thu.o.ng R[x]/(x2−x+ 1) l`a mˆo.t v´o.iR l`a tru.`o.ng c´ac sˆo´ thu.. c. T`u. d¯´o suy ra v`anh thu.o.ng R[x]/(x2−x+ 1) l`a mˆo.t tru.`o.ng.
33. X´et v`anh C c´ac sˆo´ ph´u.c v`a a =
√
2 2 +i
√
2
2 ∈C. Ch´u.ng to’ r˘a`ng tˆa.p ho..p
S = {m+na+pa2+qa3 | m, n, p, q ∈Z}
l`a v`anh con cu’a C sinh bo.’ i a. S c´o l`a mˆo.t id¯ˆean cu’a C khˆong?
34. Cho miˆ` n nguyˆene D c´o d¯o.n vi. 1 v`a 1 c´o cˆa´p n. Ch´u.ng to’ r˘a`ng:
a) n l`a mˆo.t sˆo´ nguyˆen tˆo´.
TRA’ L `O.I V `A HU.´O.NG DˆA˜N GIA’I B`AI TˆA. P
CHU.O.NG II – V `ANH
1. Ta c´o hiˆe.u d¯ˆo´i x´u.ng A+B = (A\B)∪(B\A) v`aa) A+A = ∅, b) A+∅= A, a) A+A = ∅, b) A+∅= A,
c) A+B = B+A, d) A+B = (A∪B)\(A ∩B),
e) (A+B) +C = A+ (B+C), f) A(B+C) =AB +AC.
Go.i p, q, r tu.o.ng ´u.ng l`a c´ac mˆe.nh d¯ˆe` x∈A, x ∈ B, x∈ C. Khi d¯´ox ∈A+B
ch´ınh l`a mˆe.nh d¯ˆe` tuyˆe’n loa.i (XOR) p⊕q. Ba’ng gi´a tri. chˆan l´y sau cho c´ac kˆe´t qua’ cˆau d) t`u. cˆo.t 6 v`a 7, cˆau e) t`u. cˆo.t 8 v`a 10, cˆau f) t`u. cˆo.t 11 v`a 13.
Ta c`on c´o: g) AB = BA, h) (AB)C = A(BC), i) AS =A, p q r p∨q p∧q (p∨q)∧(p∧q) p⊕q 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 1 0 1 1 1 1 0 1 1 0 0 1 1 1 1 1 0 0 (p⊕q)⊕r q⊕r p⊕(q⊕r) p∧(q⊕r) p∧r (p∧q)⊕(p∧r) 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 0
Vˆa.y P(S) v´o.i ph´ep cˆo.ng (hiˆe.u d¯ˆo´i x´u.ng) v`a ph´ep nhˆan (ph´ep giao) l`a mˆo.t v`anh giao ho´an c´o d¯o.n vi..
2. Dˆe˜ d`ang c´o d¯u.o..cZìRv´o.i ph´ep cˆo.ng l`a mˆo.t nh´om aben. Ph´ep nhˆan trˆenZìR
c´o t´ınh kˆe´t ho.. p v`a phˆan phˆo´i d¯ˆo´i v´o.i ph´ep cˆo.ng. Thˆa.t vˆa.y,∀(m, x),(n, y),(p, z) ∈
ZìR,
((m, x)(n, y))(p, z) = (mn, my +nx+xy)(p, z)
= (mnp, mnz +pmy+pnx +pxy+myz+nxz+xyz)
= (mnp, mnz +mpy+myz +npx+nxz+pxy +xyz)
= (m, x)(np, nz +py+yz) = (m, x)((n, y)(p, z)), (m, x)((n, y) + (p, z)) = (m, x)(n+p, y+z) = (mn+mp, my+mz+nx+px+xy+xz) = (mn, my +nx+xy) + (mp, mz +px+xz) = (m, x)(n, y) + (m, x)(p, z).
Ngo`ai ra ZìR c´o phˆ` n tu.a ’ d¯o.n vi. l`a (1,0). Do d¯´o ZìR l`a mˆo.t v`anh c´o d¯o.n vi.. D- ˘a.t I ={(0, x)∈ZìR} th`ıI l`a mˆo.t id¯ˆean cu’a ZìR. X´et ´anh xa.
f :R−→ ZìR : x7→ (0, x).
R˜o r`ang f l`a mˆo.t d¯o.n ´anh. Ngo`ai ra, ∀x, y ∈ R,
f(x+y) = (0, x+y) = (0, x) + (0, y) =f(x) +f(y), f(xy) = (0, xy) = (0, x)(0, y) = f(x)f(y).
Vˆa.y f l`a mˆo.t d¯o.n cˆa´u, ngh˜ıa l`a ta c´o d¯˘a’ng cˆa´u v`anh R∼= Imf = I.
3. V`ıR l`a mˆo.t v`anh v´o.i phˆa`n tu.’ khˆong l`a 0R nˆen ∀a, b, c ∈S,* a+b = f(f−1(a) +f−1(b)) =f(f−1(b) +f−1(a)) =b+a * a+b = f(f−1(a) +f−1(b)) =f(f−1(b) +f−1(a)) =b+a * (a+b) +c =f(f−1(a+b) +f−1(c)) = f(f−1(f(f−1(a) +f−1(b))) +f−1(c)) =f(f−1(a) +f−1(b) +f−1(c)) = f(f−1(a) +f−1(f(f−1(b) +f−1(c)))) =f(f−1(a) +f−1(b+c)) = a+ (b+c) * v´o.i 0S = f(0R), a+ 0S = f(f−1(a) +f−1(0S)) = f(f−1(a) + 0R) =f(f−1(a)) = a * v´o.i −a= f(−f−1(a)), a+ (−a) =f(f−1(a) +f−1(f(−f−1(a)))) = f(f−1(a) + (−f−1(a))) =f(0R) = 0S * (ab)c = f(f−1(ab)f−1(c)) =f(f−1(f(f−1(a)f−1(b)))f−1(c))
= f(f−1(a)f−1(b)f−1(c)) =f(f−1(a)f−1(f(f−1(b)f−1(c))) = f(f−1(a)f−1(bc)) =a(bc) * a(b+c) =f(f−1(a)f−1(b+c)) =f(f−1(a)f−1(f(f−1(b) +f−1(c)))) = f(f−1(a)(f−1(b) +f−1(c))) =f(f−1(a)f−1(b) +f−1(a)f−1(c)) = f(f−1(f(f−1(a)f−1(b))) +f−1(f(f−1(a)f−1(c))) = f(f−1(ab) +f−1(ac)) =ab+ac. Tu.o.ng tu.. (b+c)a =ba+ca
Vˆa.y S l`a mˆo.t v`anh. Do η l`a mˆo.t song ´anh v`a f(x + y) = f(x) + f(y),
f(xy) =f(x)f(y), ∀x, y ∈ R nˆen f l`a mˆo.t d¯˘a’ng cˆa´u.
Bˆay gi`o., nˆe´u R l`a v`anh c´o d¯o.n vi. 1 th`ı v´o.i song ´anh f : R −→ R cho bo.’ i
f(a) = 1−a (khi d¯´o f−1(a) = 1−a), R c˜ung l`a v`anh v´o.i hai ph´ep to´an
a⊕b = f−1(f(a) +f(b)) = 1−(1−a+ 1−b) = a+b−1 ab=f−1(f(a)f(b)) = 1−((1−a)(1−b)) =a+b−ab. 4. a) ∀x ∈R, 0x= x0(= 0) hay 0∈ Z(R), nˆen Z(R)6= ∅. ∀a, b ∈Z(R), (a−b)x= ax−bx =xa−xb = x(a−b), ∀x ∈R nˆen a−b ∈Z(R). (ab)x =a(bx) = a(xb) = (ax)b = (xa)b= x(ab), ∀x∈ R nˆen ab∈Z(R). R˜o r`ang ab= ba, ∀a, b ∈Z(R).
Vˆa.y Z(R) l`a mˆo.t v`anh con giao ho´an cu’a R.
V´o.i gia’ thiˆe´t R l`a mˆo.t thˆe’, ∀x ∈ R, 1x= x1 (= x), do d¯´o 1 ∈ Z(R); ngo`ai ra, ∀a ∈ Z(R), a 6= 0, ∃a−1 ∈R, aa−1 = 1; do xa= ax, ta c´o a−1x= xa−1 hay
a−1 ∈ Z(R). D- iˆe` u n`ay cho biˆe´t Z(R) l`a mˆo.t v`anh giao ho´an c´o d¯o.n vi. v`a mo.i phˆ` n tu.a ’ kh´ac 0 cu’a n´o d¯ˆ` u c´e o nghi.ch d¯a’o trong n´o, do d¯´o Z(R) l`a mˆo.t tru.`o.ng.
b) Z(M(3,R)) = ( ( a 0 0 0 a 0 0 0 a a ∈R ) . 5. a) V´o.i mo.i a ∈R, 2a = a+a = (a+a)2 = a2+ 2a2+a2 = 4a2 = 4a, do d¯´o 2a = 0 . Vˆa.y R c´o d¯˘a.c sˆo´ 2. T`u. d¯´o suy ra a = −a, ∀a ∈R.
b) V´o.i mo.i a, b ∈R, a+b= (a+b)2 = a2+ab+ba+b2 =a+ab+ba+b,do d¯´o ab+ba= 0 hay ab= −ba=ba. Vˆa.y R l`a mˆo.t v`anh giao ho´an. do d¯´o ab+ba= 0 hay ab= −ba=ba. Vˆa.y R l`a mˆo.t v`anh giao ho´an.
c) V´o.i mo.i a, b ∈R, ab(a+b) = a2b+ab2 =ab+ab= 2ab= 0, do d¯´o ho˘a.c
ab = 0 ho˘a.c a+b = 0. Nˆe´u ab= 0 th`ıa = 0 ho˘a.c b = 0. Trong tru.`o.ng ho.. p n`ay v`anh R chı’ c´o mˆo.t phˆa` n tu.’ 0. Nˆe´u ab6= 0 (t´u.c l`a a 6= 0 v`a b 6= 0) th`ıa+b = 0 hay b =−a= a. Khi d¯´o R chı’ c´o hai phˆ` n tu.a ’ .
6. a) Gia’ su.’ a v`a b lˆ` n lu.o.a . t l`a phˆ` n tu.a ’ nghi.ch d¯a’o cu’a xy v`a yx, ngh˜ıa l`a
D- ˘a.t x0 = by, x00 = ya, y0 = ax, y00 = xb th`ıx0x = xx00 = 1 v`a y0y = yy00 = 1. Do d¯´o x0 = x00 v`a y0 =y00 lˆ` n lu.o.a . t l`a phˆ` n tu.a ’ nghi.ch d¯a’o cu’a x v`a y.
b) Gia’ su.’ a l`a phˆ` n tu.a ’ nghi.ch d¯a’o cu’a xy, ngh˜ıa l`a a(xy) = (xy)a = 1. Tac´o x 6= 0 v`a y 6= 0, v`ı nˆe´u x = 0 hay y = 0 th`ıxy = 0 nˆen xy khˆong c´o nghi.ch c´o x 6= 0 v`a y 6= 0, v`ı nˆe´u x = 0 hay y = 0 th`ıxy = 0 nˆen xy khˆong c´o nghi.ch d¯a’o. D- ˘a.t x0 =ya v`a y0 =ax th`ıxx0 = y0y = 1. Khi d¯´o
x(x0x−1) =xx0x−x= 1x−x = 0 ⇒ x0x−1 = 0 ⇒ x0x = 1,
do R khˆong c´o u.´o.c cu’a khˆong v`a x = 0. Vˆ6 a.y x0 l`a phˆ` n tu.a ’ nghi.ch d¯a’o cu’a x. Tu.o.ng tu.. y0 l`a phˆ` n tu.a ’ nghi.ch d¯a’o cu’a y.
7. a) Gia’ su.’ ba = 1. Khi d¯´o nˆe´u ac= 0 th`ıc = (ba)c = b(ac) = b0 = 0. Do d¯´o
a khˆong l`a u.´o.c cu’a 0 bˆen tr´ai.
Gia’ su.’ a = ara v´o.i r ∈ R v`a a khˆong l`a u.´o.c cu’a 0 bˆen tr´ai. Khi d¯´o
a(1−ra) =a−ara = 0 nˆen ta c´o 1−ra = 0 hay ra = 1. Do d¯´o a c´o nghi.ch d¯a’o tr´ai l`a r.
b)Nˆe´u tˆ`n ta.io c ∈Rsao choc(1−ba) = 1 th`ıc(1−ba)b =bhaycb(1−ab) = b.Khi d¯´o 1 = ab+ (1 −ab) = acb(1−ab) + (1− ab) = (acb+ 1)(1 −ab). Do d¯´o Khi d¯´o 1 = ab+ (1 −ab) = acb(1−ab) + (1− ab) = (acb+ 1)(1 −ab). Do d¯´o 1−ab kha’ nghi.ch tr´ai.
8. a) V´o.i a ∈R, a6= 0, x´et ´anh xa.
fa : R−→ R : x 7→ ax.
fa l`a mˆo.t d¯o.n ´anh. Thˆa.t vˆa.y ∀x, y ∈ R, ax = ay k´eo theo a(x−y) = 0 nˆen
x−y = 0 v`ıR l`a v`anh khˆong c´o u.´o.c cu’a khˆong v`a a6= 0. Do R l`a h˜u.u ha.n nˆen
fa l`a mˆo.t song ´anh. V`ı vˆa.y, v´o.i a ∈R tˆ`n ta.io e ∈ R sao cho fa(e) = ae =a. Ta ch´u.ng minh e l`a d¯o.n vi. cu’a R.
∀x ∈ R, a(ex− x) = (ae)x−ax = ax−ax = 0, v`ıa 6= 0 nˆen ex−x = 0 hay ex = x. T`u. d¯´o ea = a v`a (xe−x)a = x(ea)−xa = xa−xa = 0, do d¯´o
xe−x = 0 hay xe =x.
V`ıfa l`a song ´anh nˆen v´o.ie ∈R, tˆ`n ta.io a0 ∈R sao chofa(a0) =aa0 = e. Ta c´o a(a0a−e) = (aa0)a−ae =ea−ae = 0 nˆen a0a =e. Vˆa.y a0 l`a phˆ` n tu.a ’ nghi.ch d¯a’o cu’a a.
b) Gia’ su.’ a c´o nghi.ch d¯a’o tr´ai l`a a0, ngh˜ıa l`a a0a= e. X´et ´anh xa.
fa : R−→ R : x 7→ ax.
fa l`a mˆo.t d¯o.n ´anh. Thˆa.t vˆa.y ∀x, y ∈ R, ax = ay k´eo theo a0(ax) = a0(ay), do d¯´o x = y. Do R l`a h˜u.u ha.n nˆen fa l`a mˆo.t song ´anh. Khi d¯´o v´o.i d¯o.n vi. e cu’a
R, tˆ`n ta.io a00 ∈ R sao cho fa(a00) = aa00 = e, t´u.c l`a a c´o nghi.ch d¯a’o pha’i l`a a00. Tu.o.ng tu.. nˆe´u a c´o nghi.ch d¯a’o pha’i th`ıa c´o nghi.ch d¯a’o tr´ai nˆen a kha’ nghi.ch.
9. a) Tˆ`n ta.io n, p∈ N∗ so cho xn = yp = 0. Theo cˆong th´u.c nhi. th´u.c Newton:(x+y)n+p−1 = (x+y)n+p−1 = n+Xp−1 k=0 Cnk+p−1xkyn+p−1−k = ( n−1 X k=0 Cnk+p−1xkyn−1−k)yp −xn( n+Xp−1 k=n Cnk+p−1xk−nyn+p−1−k) = 0. Do d¯´o x+y l`a l˜uy linh. b) Nˆe´u xn = 0 th`ı (xy)n =xnyn = 0. Do d¯´o xy l`a l˜uy linh.