Questions and Answers of Algebra Graduate Texts In Mathematics

If A is an abelian group and n > 0 an integer such that na = 0 for all a ϵ A, then A is a unitary Zn-module, with the action of Zn on A given by K̅a = ka, where k ϵ Z and k|→ K̅ ϵ Zn under
The following conditions on a ring R [with identity] are equivalent:(a) Every [unitary] R-module is projective.(b) Every short exact sequence of [unitary] R-modules is split exact.(c) Every [unitary]
(a) For any abelian group A and positive integer m, Hom(Zm,A) ≅ A[m] = {a ϵ A| ma = 0}(b) Hom(Zm,Zn) ≅ Z(m,n)•(c) The Z-module Zm has Zm * = 0.(d) For each k ≥ 1, Zm is a Zmk•module
If R = Z, then condition (iii) of Definition 5.1 is superfluous (that is, (i) and (ii) imply (iii)). Data from in Definition 5.1
(a) A set of vectors { x1, ... , Xn) in a vector space V over a division ring R is linearly dependent if and only if some xk is a linear combination of the preceding xi.(b) If { x1,X2,x3} is a
Suppose that R is a nonzero commutative ring with identity and every submodule of every free R-module is free, then R is a principal ideal domain. we need to show that every ideal of R is
If A and B are unitary K-modules [resp. K-algebras], then there is an isomorphism of K-modules [resp. K-algebras] α : A ⊗K B → B ⊗K A such that α(a⊗b) = b⊗a for all a ϵ A,b ϵ B.
Let ∫: A → B be an R-module homomorphism.(a) ∫ is a monomorphism if and only if for every pair of R-module homomorphisms g,h: D → A such that ∫g = ∫h, we have g = h. (b) ∫ is an
Let R be a ring with identity. An R-module A is injective if and only if for every left ideal L of Rand R-module homomorphism g : L → A, there exists a ϵ A such that g(r) = ra for every r ϵ L.
If A, B are abelian groups and m, n integers such that mA = 0 = nB, then every element of Hom(A,B) has order dividing (m, n).
Let A and B be abelian groups.(a) For each m > 0, A⊗Zm ≅ A/mA.(b) Zm ⊗Zn ≅ Zc, where c = (m,n).(c) Describe A⊗B, when A and Bare finitely generated.
Let A be a ring with identity. Then A is a K-algebra with identity if and only if there is a ring homomorphism of K into the center of A such that 1K|→ 1A,
Every vector space over a division ring Dis both a projective and an injective D-module. [See Exercise 1.]Data from exercise 1The following conditions on a ring R [with identity] are equivalent:(a)
Let π : Z → Z2 be the canonical epimorphism. The induced map π̅: Hom(Z2,Z) - Hom(Z2,Z2) is the zero map. Since Hom(Z2,Z2) ≠ 0 (Exercise 1(b)), π̅ is not an epimorphism.Data from exercise
If A is a torsion abelian group and Q the (additive) group of rationals, then(a) A⊗Q = 0.(b) Q⊗Q ≅ Q.
Let S be a nonempty subset of a vector space V over a division ring. The annihilator of S is the subset S0 of V* given by S° = {∫ ϵ V*| (s,∫) = 0 for all s ϵ S}.(a) 0° = V*; v0 = O; S ≠ {0}
Let A be a cyclic R-module of order r ϵ R.(a) If s ϵ R is relatively prime to r, then sA = A and A[s] = 0.(b) If s divides r, say sk = r, then sA ≅ R/(k) and A[s] ≅ R/(s).
Let A be a one-dimensional vector space over the rational field Q. If we define ab = 0 for all a,b ϵ A, then A is a Q-algebra. Every proper additive subgroup of A is an ideal of the ring A, but not
Letbe a short exact sequence of left R-modules and D a right R-module. Then1s a short exact sequence of abelian groups under any one of the following hypotheses: (a) (b) R has an identity and Dis a
If R has an identity, then every unitary cyclic R-module is isomorphic to an R-module of the form R/J, where J is a left ideal of R.
(a) If A and Bare R-modules, then the set HomR(A,B) of all R-module homomorphisms A → B is an abelian group with ∫ + g given on a ϵ A by(∫+ g)(a) = f(a) + g(a) ϵ B. The identity element is
(a) For each prime p, Z(p∞) is a divisible group.(b) No nonzero finite abelian group is divisible.(c) No nonzero free abelian group is divisible.(d) Q is a divisible abelian group.
Let R,S be rings and AR, sBR, sCR, DR (bi)modules as indicated. Let HomR denote all right R-module homomorphisms.(a) HomR(A,B) is a left S-module, with the action of S given by (s∫)(a) =
Give examples to show that each of the following may actually occur for suitable rings R and modules AR, RB.(a) A⊗RB≠ A⊗zB.(b) u ϵ A⊗R B, but u ≠ a⊗b for any a ϵ A, b ϵ B.(c) a⊗b =
Let R be a principal ideal domain, A a unitary left R-module, and p ϵ R a prime ( =_irreducible). Let pA = {pa |a ϵ A} and A[P] = {a ϵ A| pa= 0}.(a) R/(p) is a field.(b) pA and A[p] are submodules
(a) If A is a module over a commutative ring Rand a ϵ A, is an ideal of R. If a is said to be a torsion element of A.(b) If R is an integral domain, then the set T(A) of all torsion elements of
If A is a cyclic R-module of order r, then (i) every submodule of A is cyclic, with order dividing r; (ii) for every ideal (s) containing (r), A has exactly one submodule, which is cyclic of orders.
Let ℓ be the category of Exercise 1. If X is the set {x1, ••• , xn}, then the polynomial algebra K[x1, ... , xn] is a free object on the set X in the category ℓ Data from
If R has an identity, then a nonzero unitary R-module A is simple if its only submodules are O and A. (a) Every simple R-module is cyclic. (b) If A is simple every R-module endomorphism is either the
Let R be a ring with identity; then there is a ring isomorphism HomR(R,R) ≅ R0P where HomR denotes left R-module homomorphisms). In particular, if R is commutative, then there is a ring
If A' is a submodule of the right R-module A and B' is a submodule of the left R-module B, then A/ A'⊗R B/B' ≅ (A⊗R B)/C, where C is the subgroup of A ⊗R B generated by all elements a' ⊗ b
If A is a finitely generated torsion module, then {re RI rA = OI is a nonzero ideal in R, say (r1). r1 is called the minimal annihilator of A. Let A be a finite abelian group with minimal annihilator
A finitely generated R-module need not be finitely generated as an abelian group.
If G is an abelian group, then G = D⊕N, with D divisible and N reduced (meaning that N has no nontrivial divisible subgroups).
Let R and C be the fields of real and complex numbers respectively.(a) dimRC = 2 and dimRR = 1.(b) There is no field K such that R ⊂ K ⊂ C.
If A and B are cyclic modules over R of nonzero orders r and s respectively, and r is not relatively prime to s, then the invariant factors of A⊕B are the greatest common divisor of r,s and the
If ∫: A → A is an R-module homomorphism such that ∫∫= ∫, then A = Ker f Im f.
Without using Lemma 3.9 prove that:(a) Every homomorphic image of a divisible abelian group is divisible.(b) Every direct summand of a divisible abelian group is divisible.(c) A direct sum of
If Vis a vector space over a division ring and fa V*, let W = I a a V I (a,f) = 0 l, then Wis a subspace of V. If dim V is finite, what is dim W?
The usual injection α : Z2 → Z4 is a monomorphism of abelian groups. Show that l⊗α :Z2⊗Z2 → Z2⊗Z4d is the zero map (butZ2⊗Z2 ≠ O; see Exercise 2).Data from exrecise 2Let A
Proof that If G is a nontrivial group that is not cyclic of order 2, then G has a nonidentity automorphism.
Let A and a e A satisfy the hypotheses of Lemma 6.8.(a) Every R-submodule of A is an R/(pn)-module with (r + (pn)a = ra. Conversely, every R/(pn)-submodule of A is an R-submodule by pullback along R
Every torsion-free divisible abelian group D is a direct sum of copies of the rationals Q. if O ≠ n e Z and α ϵ D, then there exists a unique b ϵ D such that nb = a. Denote h by (1/n)a. For m, n
Proof that If R has an identity and we denote the left R-module R by nR and the right R-module R by RR, then (RR)*,,..__, RR and (RR)*,...._, nR.
Let be a commutative diagram of R-modules and R-module homomorphisms, with exact rows. Prove that:(a) α1 an epimorphism and α2,α4 monomorphisms => α3 is a monomorphism;(b) α5 a monomorphism
(a) of modules, then the sequence is exact.(b) Show that every exact sequence may be obtained by splicing together suitable short exact sequences as in (a). If 0 A B C 0 and 0→CD →E→0
If V is a finite dimensional vector space and vm is the vector space V⊕ V⊕· · ·⊕ V (m summands), then for each m ≥ 1, vm is finite dimensional and dim Vm = m(dim V)
If R has an identity and P is a finitely generated projective unitary left R-module, then(a) P* is a finitely generated projective right R-module.(b) P is reflexive. This proposition may be false if
Proof that:(a) If D is an abelian group with torsion subgroup Dt, then D/ Dt is torsion free.(b) If D is divisible, then so is Dt, whence D = Dt⊕E, with E torsion free.
(a) If ∫ is a right ideal of a ring R with identity and Ba left R-module, then there is a group isomorphism R/ I⊗R B ≅ B/ IB, where IB is the subgroup of B generated by all elements rb with r
If F1 and F2 are free modules over a ring with the invariant dimension property, then rank (F1 ⊕ F2) = rank F1 + rank F2.
Let A,A1, ... , An be R-modules. Then A ≅ A1 ⊕ · · · ⊕ An if and only if for each i = 1,2, ... , n there is an R-module homomorphism ,φi : A → A such that Im φi ≅ Ai; φiφj = 0
Proof that Let p be a prime and D a divisible abelian p-group. Then D is a direct sum of copies of Z(p∞).
If R,S are rings, AR, RBs, sC are (bi)modules and D an abelian group, define a middle linear map to be a function ∫: A X B X C → D such that(i) ∫(a + a',b,c) = ∫(a,b,c) +∫(a',b,c);(ii)
Let R be a ring with no zero divisors such that for all r,s ϵ R there exist a, b ϵ R, not both zero, with ar + bs = 0.(a) If R = K⊕L (module direct sum), then K = 0 or L = 0.(b) If R has an
Every divisible abelian group is a direct sum of copies of the rationals Q and copies of Z(p∞) for various primes p.
Let R be a ring with identity. Show that R is not a free module on any set in the category of all R-modules (as defined in Exercise 2).Data from in exercise 2Let R be any ring (possibly without
Let A, B, C be modules over a commutative ring R.(a) The set £(A,B;C) of all R-bilinear maps A X B → C is an R-module with (∫ + g)(a,b) = ∫(a,b) + g(a,b) and (r∫)(a,b) = r∫(a,b).(b) Each
Let F be a free module of infinite rank a over a ring R that has the invariant dimension property. For each cardinal β such that O ≤ β ≤ α, F has infinitely many proper free submodules of rank
Let G,H,K be divisible abelian groups.(a) If G ⊕ G ≅ H⊕H, then G ≅ H [see Exercise 11].(b) If G⊕H ≅ G⊕K, then H ≅ K [see Exercises 11.].Data from exercise 11Every divisible abelian
Let F be a field, X an infinite set, and V the free left F-module (vector space) on the set X. Let Fx be the set of all functions. ∫: X → F.(a) Fx is a (right) vector space over F (with ( ∫ +
Assume R has an identity. Let ϱ be the category of all unitary R-R bimodules and bimodule homomorphisms (that is, group homomorphisms ∫ : A → B such that ∫(ras) = r∫(a)s for all r,s ϵ R).
If F is a free module over a ring with identity such that F has a basis of finite cardinality n ≥ 1 and another basis of cardinality n + 1, then F has a basis of cardinality m for every m ≥ n (m
Proof that Let K be a ring with identity and Fa free K-module with an infinite denumerable basis {e1,e2, ... }. Then R = HomK(F,F) is a ring by Exercise 1. 7(b). If n is any positive integer, then
Show that isomorphism of short exact sequences is an equivalence relation.
If D is a ring with identity such that every unitary D-module is free, then D is a division ring.
Let ∫: V → V' be a linear transformation of finite dimensional vector spaces V and V' such that dim V = dim V'. Then the following conditions are equivalent: (i) ∫ is an isomorphism; (ii) ∫
If ∫: A → B and g : B → A are R-module homomorphisms such that g∫ = 1A, then B = Im ∫⊕ Ker g.
Let R be a ring and R0P its opposite ring. If A is a left [resp. right] R-module, then A is a right [resp. left] R0P-module such that ra = ar for all a ϵ A, r ϵ R, r ϵ R0P.
{a) If R has an identity and A is an R-module, then there are submodules Band C of A such that B is unitary, RC = 0 and A = B ⊕ C. (b) Let A1 be another R-module, with A1 = B1⊕C1 (B1
Consider the set G = {±1, ±i, ±j, ±k} with multiplication given by i2 = j2 = k2 = -1; ij = k; jk = i, ki =j; ji = -k, kj = -i, ik = -j, and the usual rules for multiplying by ± 1. Show that G is
Let G be a finite abelian group and x an element of maximal order. Show that (x) is a direct summand of G. Use this to obtain another proof of Theorem 2.1.Data from theorem 2.1Every finitely
Give examples other than those in the text of semigroups and monoids that are not groups.
(a) If G is an abelian group and m ϵ Z, then mG = { mu| u ϵ G} is a subgroup of G.(b) If G=G₁, then mGmG₁ and G/mG= =4 iel
A nonempty finite subset of a group is a subgroup if and only if it is closed under the product in G.
Let G be a group and a,b ϵ G such that (i) lal = 4 = lbl; (ii) a2 = b2 ; (iii) ba = a3b = a-1b; (iv) a ≠ b; (v) G = (a,b). Show that IGI = 8 and G ≅ Q8.
If N ⊲ G and N, G/ N are both p-groups, then G is a p-group.
A group G is indecomposable if and only if G ≠ (e) and G ≅ H X K implies H = ( e) or K = ( e).
Let G and H be groups and θ : H → Aut Ga homomorphism. Let G Xθ H be the set G X H with the following binary operation: (g,h)(g',h') = (g[θ(h)(g')],hh'). Show that G Xθ H is a group with
If G = Go > G1 > · · · > Gn is a subnormal series of a finite group G, then |G| = 'n-1 II|G₂/G₁+1||G₂|.
{a) A4 is not the direct product of its Sylow subgroups, but A4 does have the property: mn = 12 and (m,n) = 1 imply there is a subgroup of order m.(b) S3 has subgroups of orders 1, 2, 3, and 6 but is
(a) Find a normal series of D4 consisting of 4 subgroups.(b) Find all composition series of the group D4•(c) Do part (b) for the group A4•(d) Do part (b) for the group S3 X Z2.(e) Find all
Let G be a group and A a normal abelian subgroup. Show that G / A operates on A by conjugation and obtain a homomorphism G/ A → Aut A.
Proof that If G is a finite p-group, H ⊲ G and H ≠ (e), then H ∩ C(G) ≠ (e).
Let G be a group and a,b ϵ G. Denote the commutator aba-1b-1ϵ G by [a,b]. Show that for any a,b,c, ϵ G, [ab,c] = a[b,c]a- 1[a,c].
A subset X of an abelian group F is said to be linearly independent if n1x1 + · · · + nkxk = 0 always implies ni = 0 for all i (where n, ϵ Z and x1, ... , xk are distinct elements of X).(a) X is
If H,K are subgroups of G such that H ⊲ K show that K < NG(H).
Proof that Let |G| = pn. For each k, 0 ≤ k ≤ n, G has a normal subgroup of order pk.
If H and K are subgroups of a group G, let (H,K) be the subgroup of G generated by the elements { hkh-1k1| h ϵ H, k ϵ K}. Show that{a) (H,K) is normal in H V K.(b) If (H,G') = (e), then (H',G) =
If N is a simple normal subgroup of a group G and G/ N has a composition series, then G has a composition series.
Proof that Let X = { ai|i ϵ I| be a set. Then the free abelian group on X is (isomorphic to) the group defined by the generators X and the relations (in multiplicative notation) { aiajai-1aj-1 = e|
Proof that If a group G contains an element a having exactly two conjugates, then G has a proper normal subgroup N ≠ (e).
Let H be a subgroup of G. The centralizer of H is the set CG(H) = {g ϵ G| hg = gh for all h ϵ H}. Show that CG(H) is a subgroup of NG (H).
If G is an infinite p-group (p prime), then either G has a subgroup of order pn for each n ≥ 1 or there exists m ϵ N* such that every finite subgroup of G has order ≤ pm.
Proof that A nontrivial homomorphic image of an indecomposable group need not be indecomposable.
What i& the center of the quaternion group Q8? Show that Q8/C(Q8) is abelian.
Define a chain of subgroups ϒi,{G) of a group G as follows: 'ϒ1(G) = G, ϒ2{G) = (G,G), ϒi,{G) = {ϒi_1(G),G) (see Exercise 3). Show that G is nilpotent if and only if ϒm{G) = (e) for some m.Data
Proof that A composition series of a group is a subnormal series of maximal (finite) length.
If H is a subgroup of G, the factor group NG(H)/CG(H) (see Exercise 4) is isomorphic to a subgroup of Aut H.Data From Exercise 4Let H be a subgroup of G. The centralizer of H is the set CG(H) = {g ϵ
Proof that If P is a normal Sylow p-subgroup of a finite group G and ∫ : G → G is an endomorphism, then ∫(P) < P.

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