Questions and Answers of Algebra Graduate Texts In Mathematics

If H is a maximal proper subgroup of a finite solvable group G, then [G: HJ is a prime power.
Proof that for any group G, C(G) is characteristic, but not necessarily fully invariant.
For each x ϵ S, Gx is a maximal (proper) subgroup of G. The proof of this fact proceeds in several steps:(a) A block of G is a subset T of S such that for each g ϵ G either gT ∩ T = Ø or gT = T,
(a) If φ : R → S is a homomorphism of rings, then the map φ̅ : R[[x]] → S[[x]] given by is a homomorphism of rings such that φ̅ (R[x]) ⊂ S[x]. (b) φ̅ is a monomorphism [epimorphism]
Let |Ri| i ϵ I} be a family of rings with identity. Make the direct sum of abelian groups into a ring by defining multiplication coordinate wise. Does have an identity? Σ R iel
Proof that If G is a finite nilpotent group, then every minimal normal subgroup of G is contained in C( G) and has prime order.
(a) Let S be a nonempty set and let Ns be the set of all functions φ : S → N such that φ(s) ≠ O for at most a finite number of elements s ϵ S. Then Ns is a multiplicative abelian monoid with
If F is a field and ∫,g ϵ F[x] with deg g ≥ 1, then there exist unique polynomials ∫o,∫1, ... ,∫ϒ ϵ F[x] such that deg ∫i < deg g for all i and f = fo + fig + f28²+...+
Proof that A nonzero ideal in a principal ideal domain is maximal if and only if it is prime.
The set of all nilpotent elements in a commutative ring forms an ideal.
Determine the complete ring of quotients of the ring zn for each n ≥ 2.
(a) If D is an integral domain and c is an irreducible element in D, then D[x] is not a principal ideal domain.(b) Z[x] is not a principal ideal domain.(c) If F is a field and n ≥ 2, then F[x1, ...
(a) Let G be an (additive) abelian group. Define an operation of multiplication in G by ab = 0 (for all a,b ϵ G). Then G is a ring.(b) Let S be the set of all subsets of some fixed set U. For A,B ϵ
An integral domain R is a unique factorization domain if and only if every nonzero prime ideal in R contains a nonzero principal ideal that is prime.
Let I be an ideal in a commutative ring R and let Rad I = {r ϵ R| rn ϵ I for some n}. Show that Rad I is an ideal.
Let S be a multiplicative subset of a commutative ring R with identity and let T be a multiplicative subset of the ring s-1R. Let S* = {r ϵ R| r/ s ϵ T for some s ϵ S}. Then S* is a multiplicative
Let MatnR be the ring of n X n matrices over a ring R. Then for each n ≥ 1:(a) (MatnR)[x] ≅ MatnR[x].(b) (MatnR)[[x]]"' Mat,.R[[x]].
If R is a ring and a ϵ R, then J = {r ϵ R| ra = 0} is a left ideal and K = {r ϵ R|ra = 0| is a right ideal in R.
(a) The set E of positive even integers is a multiplicative subset of Z such that E-1(Z) is the field of rational numbers.(b) State and prove condition(s) on a multiplicative subset S of Z which
Let∫ be a polynomial of positive degree over an integral domain D.(a) If char D = 0, then ∫' ≠ 0.(b) If char D = p ≠ 0, then ∫' = 0 if and only if /is a polynomial in xP (that is, ∫ = a0
A ring R such that a2 = a for all a ϵ R is called a Boolean ring. Prove that every Boolean ring R is commutative and a + a = 0 for all a ϵ R. [For an example of a Boolean ring, see Exercise
Let R be a ring and Gan infinite multiplicative cyclic group with generator denoted x. Is the group ring R(G) isomorphic to the polynomial ring in one indeterminate over R?
Show that in the integral domain of Exercise 3 every element can be factored into a product of irreducible, but this factorization need not be unique (in the sense of Definition 3.5 (ii)). In
(a) The center of the ring S of all 2 X 2 matrices over a field F consists of all matrices of the form (b) The center of Sis not an ideal in S.(c) What is the center of the ring of all n X n
If I is a left ideal of R, then A(I) = {r ϵ R| rx = 0 for every x ϵ I} is an ideal in R.
Proof that If D is a unique factorization domain, a ϵ D and ∫ ϵ D[x], then C(a∫) and aC(∫) are associates in D.
Let R be a ring and Sa nonempty set. Then the group M(S,R) is a ring with multiplication defined as follows: the product of ∫,g ϵ M(S,R) is the function S → R given by s|→ ∫(s) g (s).
Let R and S be rings with identity, φ : R → S a homomorphism of rings with φ(1R) = 1s, and s1,s2, ••• , sn ϵ S such that Sisj = siSj for all i,j and φ(r)si = Siφ(r) for all r ϵ R and
Let R be a principal ideal domain.(a) Every proper ideal is a product P1P2· • •Pn of maximal ideals, which are uniquely determined up to order.(b) An ideal P in R is said to be primary if ab ϵ
If I is an ideal in a ring R, let [R: I] = {r ϵ R|xr ϵ I for every x ϵ R}. Prove that [R : I] is an ideal of R which contains I.
Proof that Let R be an integral domain with quotient field F. lf T is an integral domain such that R ⊂ T ⊂ F, then F is (isomorphic to) the quotient field of T.
Let R be the following subring of the complex numbers: Then R is a principal ideal domain that is not a Euclidean domain. R = = {a + b(1 + √19 i)/2 a,b & Z}
Proof that If A is the abelian group Z⊕Z, then End A is a noncommutative ring.
(a) If R is the ring of all 2 X 2 matrices over Z, then for any A ϵ R, (x + A)(x - A) = x2 - A2 ϵ R[x].(b) There exist C,A ϵ R such that (C + A)(C - A) ≠ C2 - A2• Therefore, Corollary 5.6 is
Proof that:(a) If a and n are integers, n > 0, then there exist integers q and r such that a = qn + r, where |r| ≤ n/2.(b) The Gaussian integers Z[i] form a Euclidean domain with φ(a+bi) = a2+b2
Let p ϵ Z be a prime; let F be a field and let c ϵ F. Then xp - c is irreducible in F[x] if and only if xP - c has no root in F.
A finite ring with more than one element and no zero divisors is a division ring.
If R is a commutative ring with identity and ∫= anxn + • • • + a0 is a zero divisor in R[x], then there exists a nonzero b ϵ R such that ban. = ban-1 = · · ·= bao = 0.
What are the units in the ring of Gaussian integers Z[i]?
(a) A ring R with identity is a division ring if and only if R has no proper left ideals.(b) If S is a ring (possibly without identity) with no proper left ideals, then either S2 = 0 or S is a
Let R1 and R2 be integral domains with quotient fields F1 and F2 respectively. If ∫ : R1 → R2 is an isomorphism, then f extends to an isomorphism F1 ≅ F2,
(a) The polynomial x + I is a unit in the power series ring Z[[x]], but is not a unit in Z[x].(b) x2 + 3x + 2 is irreducible in Z[[x]], but not in Z[x].
If F is a field, then (x) is a maximal ideal in F[x], but it is not the only maximal ideal (compare Corollary 5.10).
Let R be a commutative ring with identity, I an ideal of R and π : R → R/ I the canonical projection.(a) If S is a multiplicative subset of R, then πS = π(S) is a multiplicative subset of R/
Let S be the ring of all n X n matrices over a division ring D.(a) S has no proper ideals (that is, 0 is a maximal ideal). 8 or argue directly, using the matrices Er,s mentioned there.](b) S has zero
If R = Z, A1 = (6) and A2 = (4), then the map θ : R/ A1 ∩ A2 → R/ A1 X R/ A2 of Corollary 2.27 is not surjective.
Let ℓ be the category whose objects are all commutative K-algebras with identity and whose morphisms are all K-algebra homomorphisms ∫: A → B such that ∫(1A) = 1B. Then any two
Let R be the set of all 2 X 2 matrices over the complex field C of the form where z̅,w̅ are the complex conjugates of z and w respectively (that is, Then Risa division ring that is isomorphic to
Let k,n be integers such that 0 ≤ k ≤ n and the binomial coefficient n!;(n - k)!k!, where 0! = 1 and for n > 0, n! = n(n - l)(n - 2)· • ·2· l. (a)(b) (c) (d) is an integer.(e) if p
Let have degree n. Suppose that for some k (0 n ; p ł ak; p| ai for all O ≤ i ≤ k - 1; and p2 ł a0• Show that ∫ has a factor g of degree at least k that is irreducible in Z[x]. [= Σ
(a) Let D be an integral domain and c ϵ D. Let ∫(x) and ∫(x - c) = Then f(x) is irreducible in D[x] if and only ∫(x - c) is irreducible.(b) For each prime p, the cyclotomic polynomial ∫ =
Let R be a ring with identity and S the ring of all n X n matrices over R. J is an ideal of S if and only if J is the ring of all n X n matrices over I for some ideal I in R.
Let R be a commutative ring with identity of prime characteristic p. If a,b ϵ R, then (a ± b)pn = apn ± bpn for all integers n ≥ 0 [see Theorem 1.6 and Exercise 10; note that b = -b if p =
(a) Let c ϵ F, where F is a field of characteristic p (p prime). Then xP - x - c is irreducible in F[x] if and only if xp - x - c has no root in F.(b) If char F = 0, part (a) is false.
Let R be a unique factorization domain and d a nonzero element of R. There are only a finite number of distinct principal ideals that contain the ideal (d).
Let S be a multiplicative subset of a commutative ring R with identity. If I is an ideal in R, then s-1(Rad I) = Rad (S-1I).
(a) The subset G = {1,-1,i,-i,i,-j,k,-k} of the division ring K of real quaternions forms a group under multiplication.(b) G is isomorphic to the quaternion group.(c) What is the difference between
If co, c1, ... , Cn are distinct elements of an integral domain D and do, ... , dn are any elements of D, then there is at most one polynomial ∫ of degree ≤ n in D[x] such that ∫(ci) = di for i
Proof that:(a) If F is a field then every nonzero element of F[[x]] is of the form xku with u ϵ F[[x]] a unit.(b) F[[x]] is a principal ideal domain whose only ideals are 0, Fl[x]] = (1F) = (xO) and
If R is a unique factorization domain and a,b ϵ R are relatively prime and a|bc, then a|c.
(a) Show that Z is a principal ideal ring. See Theorem 3.1 (b) Every homomorphic image of a principal ideal ring is also a principal ideal ring. (c) Zm is a principal ideal ring for every m >
Lagrange's Interpolation Formula. If F is a field, a0,a1, ... , an are distinct elements of F and c0,c1, ... , Cn are any elements of F, then is the unique polynomial of degree ≤ n in F[x] such
Let R be an integral domain and for each maximal ideal M (which is also prime, of course), consider RM as a subring of the quotient· field of R. Show that ∩ RM = R, where the intersection is taken
Let ℓ be the category with objects all commutative rings with identity and morphisms all ring homomorphisms ∫: R → S such that ∫{1R) = 1s. Then the polynomial ring Z[x1, ... , xn] is a
Proof that Let R be a Euclidean ring and a ϵ R. Then α is a unit in R if and only if φ(a) = φ(1R)
If N is the ideal of all nilpotent elements in a commutative ring R (see Exercise 1), then R/ N is a ring with no nonzero nilpotent elements.Data from exercise 1The set of all nilpotent elements in a
Let p be a prime in Z; then (p) is a prime ideal. What can be said about the relationship of Zp and the localization Z(p)?
Every nonempty set of elements (possibly infinite) in a commutative principal ideal ring with identity has a greatest common divisor.
A commutative ring with identity is local if and only if for all r, s ϵ R, r + s = 1R implies r or s is a unit.
An element of a ring is nilpotent if an = 0 for some n. Prove that in a commutative ring a + b is nilpotent if a and b are. Show that this result may be false if R is not commutative.
Let R be a Euclidean domain with associated function φ : R - {0} → N. a = qob +r₁, with r₁=0 b = qrr2, with r₂ = 0 92r2r3, with r3 = 0 r₁ = or (r₁) < y(b); or_y(r₂) < y(ri); or (r3)
If F is a field, then x and y are relatively prime in the polynomial domain F[x,y], but F[x,y] = # (1) + (x) 2 (41)
Let ∫: R → S be a homomorphism of rings, I an ideal in R, and Jan ideal in S.(a) ∫-1(J) is an ideal in R that contains Ker ∫.(b) If ∫ is an epimorphism, then ∫(I) is an ideal in S. If ∫
Show that the ring R consisting of all rational numbers with denominators not divisible by some (fixed) prime p is a local ring.
Let D be a unique factorization domain with a finite number of units and quotient field F. If ∫ ϵ D[x] has degree n and co,c1, ... , cn are n + 1 distinct elements of D, then ∫ is completely
In a ring R the following conditions are equivalent.(a) R has no nonzero nilpotent elements (see Exercise 12).(b) If a ϵ R and a2 = 0, then a = 0.Data from exercise 12An element of a ring is
If P is an ideal in a not necessarily commutative ring R, then the following conditions are equivalent.(a) P is a prime ideal.(b) If r,s ϵ R are such that rRs ⊂ P, then r ϵ P or s ϵ P.(c) If (r)
If M is a maximal ideal in a commutative ring R with identity and n is a positive integer, then the ring R/ Mn has a unique prime ideal and therefore is local.
Let R be a commutative ring with identity and c,d ϵ R with c a unit.(a) Show that the assignment x|→ cx + b induces a unique automorphism of R[x] that is the identity of R. What is its inverse?(b)
The set consisting of zero and all zero divisors in a commutative ring with identity contains at least one prime ideal.
In a commutative ring R with identity the following conditions are equivalent:(i) R has a unique prime ideal;(ii) every nonunit is nilpotent(iii) R has a minimal prime ideal which contains all zero
(a) Give an example of a nonzero homomorphism ∫ : R → S of rings with identity such that ∫(1R) ≠ 1s.(b) If ∫: R → S is an epimorphism of rings with identity, then ∫(1R) = 1s.(c) If ∫:
Let R be a commutative ring with identity and suppose that the ideal A of R is contained in a finite union of prime ideals P1 U • • • U Pn. Show that A ⊂ P, for some i.
Proof that Every nonzero homomorphic image of a local ring is local.
Let ∫ = anxn + · · · + a0 be a polynomial over the field R of real numbers and Jet φ = |an|xn + · · · + |ao| ϵ R[x].(a) If |u| ≤ d, then |f(u)| ≤ φ(d). [Recall that |a + b| ≤ |a| +
Let ∫: R → s be a homomorphism of rings such that ∫(r) ≠ 0 for some nonzero r ϵ R. If R has an identity and S has no zero divisors, then S is a ring with identity ∫(1R). Terminology due to
(a) If R is a ring. then so is Rop. where Rop is defined as follows. The underlying set of Rop is precisely Rand addition in Rop coincides with addition in R. Multiplication in Rop, denoted 0 , is
Let ∫ : R → S be an epimorphism of rings with kernel K.(a) If P is a prime ideal in R that contains K, then f(P) is a prime ideal in S [see Exercise 13].(b) If Q is a prime ideal in S, then
If one attempted to dualize the notion of free module over a ring R (and called the object so defined "co-free") the definition would read: An R-module F is co-free on a set X if there exists a
Let Q be the field of rational numbers and R any ring. If ∫,g : Q → R are homomorphisms of rings such that ∫|Z = g| Z, then ∫ = g.
An ideal M ≠ R in a commutative ring R with identity is maximal if and only if for every r ϵ R - M, there exists x ϵ R such that 1R - rx ϵ M.
The ring E of even integers contains a maximal ideal M such that E/ M is nor a field.
In the ring Z the following conditions on a nonzero ideal/ are equivalent:(i) I is prime;(ii) I is maximal;(iii) I = (p) with p prime.
Determine all prime and maximal ideals in the ring Zm,
(a) If R1. ... , Rn are rings with identity and I is an ideal in R1 X --· X Rn, then I = A1 X · · · X Am, where each Ai is an ideal in Ri.(b) Show that the conclusion of (a) need not hold if the
An element e in a ring R is said to be idempotent if e2 = e. An element of the center of the ring R is said to be central. If e is a central idempotent in a ring R with identity, then(a) 1R - e is a
Idempotent elements e1, ... , en in a ring R [see Exercise 23} are said to be orthogonal if eiej = 0 for i ≠ j. If R, R1. ... , Rn are rings with identity, then the following conditions are
Let R be any ring (possibly without identity) and X a nonempty set. In this exercise an R-module Fis called a free module on X if F is a free object on X in the category of a/I left R-modules. Thus
Let R be any ring (possibly without identity) and Fa free R-module on the set X, with L : X → F, as in Exercise 2. Show that L(X) is a set of generators of the R-module F. is commutative. Conclude
Proof that If m ϵ Z has a prime decomposition m = p1k1. · ·ptkt (ki > 0; pi distinct primes), then there is an isomorphism of rings Zm ≅ Zp1k1 X · · · X Zptkt.

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