Questions and Answers of Algebra Graduate Texts In Mathematics

Let ni, n2, ... , n2, nt be positive integers such that n1 + n2 + • · · + nt = n and for each i let Mi be an ni X ni matrix. Let M be then X n matrix where the main diagonal of each Mi lies
Let K be a field and A ϵ MatnK.(a) If A is nilpotent (that is, Am = 0 for some m), then Tr Ar = 0 for all r ≥ 1. (b) If char K = 0 and Tr Ar = 0 for all r ≥ 1, the~ A is nilpotent.
Given the set {a1, ... , an) and the words w1, w2, ••• , Wr ( on the ai), let F* be the free (nonabelian multiplicative) group on the set {a1, ... , an) and let M be the normal subgroup
Let F be a free abelian group with basis{a1, ... , am). Let K be the subgroup of F generated by b1 = r11a1 + • "• + r1mam ... , ... , bn = rn1a1 + • • • + rnmam (rij ϵ Z).(a) For each i,
Find all possible [primary] rational canonical forms for a matrix A ϵ MatnQ such that (i) A is 6 X 6 with minimal polynomial (x - 2)2(x + 3); (ii) A is 7 X 7 with minimal polynomial (x2 +
Determine the structure of the abelian group G defined by generators {a,b} and relations 2a + 4b = 0 and 3b = 0. Do the same for the group with generators {a,b,c,d} and relations 2a + 3b = 4a = 5c +
(a) Let R be an integral domain with quotient field F. If O ≠ a ϵ R, then the following are equivalent:(i) Every nonzero prime ideal of R contains a;(ii) Every nonzero ideal of R contains some
Let R and J be as in Exercise 4; let A be a finitely generated R-module and ∫ : C → A an R-module homomorphism. Then ∫ induces a homomorphism ∫̅: C/JC → A/JA in the usual way (Corollary
(a) If every prime ideal in an integral domain R is invertible, then R is Dedekind.(b) If R is a Noetherian integral domain in which every maximal ideal is invertible, then R is Dedekind
Let R have an identity. A prime ideal Pin R is called a minimal prime ideal of the ideal I if I ⊂ P and there is no prime ideal P' such that I ⊂ P' (a) If an ideal I of R is contained in a prime
The radical of an ideal I in a ring R with identity is the intersection of all its minimal prime ideals [see Exercise 6].Data from exercise 6Let R have an identity. A prime ideal Pin R is called a
Let R be a commutative ring with identity and let ∫,g e R[[xJJ. Denote by In ∫, the "' initial degree of ∫ (that is, the smallest n such that an ≠ 0, where Show that(a) In (∫ + g) ≥
Let R be Noetherian and let B be an R-module. If P is a prime ideal such that P = ann x for some nonzero x ϵ B (see Exercise 7), then P is called an associated prime of B.(a) If B ≠ 0, then there
Every nonempty K-variety in Fn may be written uniquely as a finite union V1 U V2 U · · · U Vk of affine K-varieties in Fn such that Vj ⊄ Vi for i ≠ j and each Vi is irreducible (Exercise
If F is a field, then:(a) The ideal (x, y) is maximal in F[x, y]; (b) (x, y)2 = (x2, xy, y2) (c) The ideal (x2,y) is primary and the only proper prime ideal containing it is (x, y). (x²,y)(x,y);
If R is a Dedekind domain with quotient field K, F is a finite dimensional extension field of K and Sis the integral closure of R in F (that is, the ring of all elements of F that are integral over
Let ∫: R → S be an epimorphism of commutative rings with identity. If J is an ideal of S, let I = ∫-1(J).(a) Then ∫ is primary in R if and only if J is primary in S.(b) If J is primary for P,
Let R and B be as in Exercise 12. Then the associated primes of B are precisely the primes P1, .. . , Pn, where O = A1 ∩ · · · ∩ An is a reduced primary decomposition of O with each
A valuation domain is an integral domain R such that for all a,b ϵ R either a I b or b I a. (Clearly a discrete valuation ring is a valuation domain.) A Prüfer domain is an integral domain in which
Find a reduced primary decomposition for the ideal I = (x2, xy, 2) in Z[x, y] and determine the associated primes of the primary ideals appearing in this decomposition.
Let S be a multiplicative subset of R and let A be a P-primary submodule of an R-module B. If p ∩ s = Ø then s-1A is an s-1P-primary submodule of the s-1R-module s-1B.
(a) If p is prime and n > 1, then (pn) is a primary, but not a prime ideal of Z.(b) Obtain a reduced primary decomposition of the ideal (12600) in z.
A subset T of R is said to be an m-system (generalized multiplicative system) if (a) P is a prime ideal of R if and only if R - P is an m-system.(b) Let I be an ideal of R that is disjoint from an
Let F be a field of characteristic O and R = F[x,y] the additive group of polynomials in two indeterminates. Define multiplication in R by requiring that multiplication be distributive, that ax = xa,
For each a, b ϵ R let a º b = a+ b + ab. (a) ° is an associative binary operation with identity element O ϵ R.(b) The set G of all elements of R that are both left and right quasi-regular
(a) In the ring Z[x], the following are primary decompositions:(4, 2x,x2) = (4, x) n (2, x2);(9,3x + 3) = (3) ∩ (9, x + 1).(b) Are the primary decompositions of part (a) reduced?
If F is a field and / is the ideal (x2, xy) in F[x, yJ, then there are at least three distinct reduced primary decompositions of I; three such are (i) I = (x) ∩ (x2, y);(ii) I= (x) ∩ (x2, x
A ring R is subdirectly irreducible if the intersection of all nonzero ideals of R is nonzero.(a) R is subdirectly irreducible if and only if whenever R is isomorphic to a subdirect product of {Ri Ii
R is a division ring if and only if every element of R except one is left quasi-regular.Data from exercise 1 For each a, b ϵ R let a º b = a+ b + ab. (a) o is an associative binary operation
(a) If M is a left algebra A-module, then α(M) = {r ϵ A I rc = 0 for all c ϵ M} is an algebra ideal of A.(b) An algebra ideal P of A is said to be primitive if the quotient algebra R/P is
Let I be a left ideal of Rand let(I: R) = {r ϵ RI rR ⊂ I}.(a) (I: R) is an 1deal of R. If I is regular, then (I: R) is the largest ideal of R that is contained in I.(b) If I is a regular maximal
(a) If I is an ideal of R, then P(I) = I ∩ P(R). In particular, P(P(R)) = P(R). (b) P(R) is the smallest ideal K of R such that P(R/K) = 0. In particular, P(R/P(R)) = 0, whence R/P(R) is semi
Let V be an infinite dimensional vector space over a division ring D.(a) If F is the set of all θ ϵ HomD(V,V) such that Im θ is finite dimensional, then F is a proper ideal of HomD(V,V). Therefore
Let M be a simple algebra A-module.(a) D = HomA(M,M) is a division algebra over K, where HomA(M,M) denotes all endomorphisms of the algebra A-module M.(b) M is a left algebra D-module.(c) The ring
Let V be a vector space over a division ring D. A subring R of HomD(V,V) is said to be n-fold transitive if for every k (1 ≤ k ≤ n) and every linearly independent subset I u1, . .. , uk} of V and
Construct functors as follows:(a) A covariant functor G → S that assigns to each group the set of all its subgroups.(b) A covariant functor R → R that assigns to each ring N the polynomial
Let (A, α) and (B, β) be representations of the covariant functors S: ℓ → S and T: ℓ → S respectively. If T: S → T is a natural transformation, then there is a unique morphism ∫: A →
(a) Let P : S → S be the functor that assigns to each set X its power set (set of all subsets) P(X) and to each function ∫: A → B the map P(∫) : P(B) → P(A) that sends a subset X of B onto
(a) The forgetful functor M → S (see the Example preceding Definition 1.2) is representable.(b) The forgetful functor G → S is representable.Data from definition 1.2 Definition 1.2. Let C
(a) Let ϕ and Ψ be endomorphisms of a finite dimensional vector space E such that ϕΨ = Ψϕ. If E has a basis of eigenvectors of ϕ and a basis of eigenvectors of Ψ then E has a basis consisting
If where a̅ is the image of a under the canonical epimorphism z → zP.(a) If ∫ is monic and /is irreducible in Zp[x] for some prime p, then f is irreducible in Z[x].(b) Give an example
Assume E and Fare the quotient fields of integral domains Rand S respectively. Then C is an R-module and an S-module in the obvious way (a) E and F are linearly disjoint over K if and only if
E and F are free over K if every subset X of E that is algebraically independent over K is also algebraically independent over F.(a) The definition is symmetric (that is, E and Fare free over K if
Let ∫, g : E → E, h : E → F, k : F → G be linear transformations of left vector spaces over a division ring D with dimDE = n, dimDF = m, dimDG = p. (a) Rank (∫ + g) ≤ rank ∫ + rank
Let R be commutative.(a) If the matrix product AB is defined, then so is the product BtAt and (AB)t= BtAt.(b) If A is invertible, then so is At and (At)-1 = (A-1)t.(c) If R is not commutative, then
A matrix (aij) ϵ MatnR is said to be Prove that the set of all diagonal matrices is a subring of MatnR which is (ring) isomorphic to R x · · · x R(n factors). Show that the set T of all
If (ci1,ci2· · · cim) is a nonzero row of a matrix (cij), then its leading entry is cit where r is the first integer such that cit ≠ 0. A matrix C = (cij) over a division ring D is said to be in
(a) The system of n linear equations in m unknowns x, over a field K has a (simultaneous) solution if and only if the matrix equation AX = B has a solution X, where A is the n X m matrix (aij),
Let K be a field and A ϵ MatnK.(a) 0 is an eigenvalue of A if and only if A is not invertible.(b) If k1, ... , kr ϵ K are the (not necessarily distinct) eigenvalues of A and ∫ ϵ K[x], then
Let R be a principal ideal domain. For each positive integer rand sequence of nonzero ideals I1 ⊃ I2 ⊃ • . • ⊃ Ir choose a sequence d1, ... , dr ϵ R such that (dj) = Ij and d1| d2 I· ·
A matrix A ϵ MatnR is symmetric if A = At and skew-symmetric if A = -At.(a) If A and B are [skew] symmetric, then A +B is [skew] symmetric.(b) Let R be commutative. If A,B are symmetric, then AB is
(a) If T: ℓ →D is a covariant functor, let Im T consist of the objects {T(C) I C ϵ ℓ} and the morphisms {T(∫) : T(C) → T(C') I ∫: C → C' a morphism in ℓ}. Then show that Im T
A commutative diagram of morphisms of a category e is called a pullback for ∫1 and ∫2 if for every pair of morphisms h1 : B'-----+ C1, h2: B' → C2 such that ∫1h1 = ∫2h2 there exists a
Let ν be the category whose objects are all finite dimensional vector spaces over a field F (of characteristic ≠ 2,3) and whose morphisms are all vector-space isomorphisms. Consider the dual space
Let J, g : C → D be morphisms of a category ℓ. For each X in ℓ let (a) Eq( - ,∫,g) is a contra variant functor from ℓ to the category of sets.(b) A morphism i: K → C is a difference
(a) Let S : ℓ → D and T : ℓ → D be covariant functors and α : S → T a natural isomorphism. Then there is a natural isomorphism β: T → S such that βα = IS and αβ = IT, where IS : S
An element a of a ring R is regular (in the sense of Von Neumann) if there exists x ϵ R such that axa = a. If every element of R is regular, then R is said to be a regular ring. (a) Every
(a) Let R be a ring with identity. The matrix ring MatnR is prime if and only if R is prime.(b) If R is any ring, then P(MatnR) = MatnP(R).
If R has an identity, then(a) J(R) = {r ϵ R I 1R + sr is left invertible for all s ϵ R}.(b) J(R) is the largest ideal K such that for all r ϵ K, 1R + r is a unit.
Let A be the set of all denumerably infinite matrices over a field K (that is, matrices with rows and columns indexed by N*) which have only a finite number of nonzero entries.(a) A is a simple
(a) Every nonzero homomorphic image and every nonzero submodule of a semi simple module is semi simple.(b) The intersection of two semi simple submodules is 0 or semi simple.
If A and B are left Artinian algebras over a field K, then A ⊗ K B need not be left Artinian. Let A be a division algebra with center K and maximal subfield B such that dimBA is infinite.
(a) The homomorphic image of a semi simple ring need not be semi simple.(b) If ∫ : R → S is a ring epimorphism, then f(J(R}) ⊂ J(S).
If R is a primitive ring with identity and e ϵ R is such that e2 = e ≠ 0, then(a) eRe is a subring of R, with identity e.(b) eRe is primitive.
The following conditions on a semi simple module A are equivalent:(a) A is finitely generated.(b) A is a direct sum of a finite number of simple submodules.(c) A has a composition series.(d) A
If D is finite dimensional division algebra over its center K and F is a maximal subfield of D, then there is a K-algebra isomorphism D⊗k F ≅ MatnF, where n = dimFD.
If R is the ring of all rational numbers with odd denominators, then J(R) consists of all rational numbers with odd denominator and even numerator.
The following are equivalent:(a) R is prime;(b) a, b ϵ R and aRb = 0 imply a = 0 or b = 0;(c) The right annihilator of every nonzero right ideal of R is 0;(d) The left annihilator of every nonzero
Let R be a ring that (as a left R-module) is the sum of its minimal left ideals. Assume that {r ϵ R I Rr = 0} = 0. If A is an R-module such that RA = A, then A is semi simple.
Let D be a division ring with center K. If a, b ϵ D are algebraic over the field K and have the same minimal polynomial, then b = dad-1 for some d ϵ D.
A principal ideal domain R is semi simple if and only if Risa field or R contains an infinite number of distinct non associate irreducible elements.
Let R be the ring of 2 × 2 matrices over an infinite field.(a) R has an infinite number of distinct proper left ideals, any two of which are isomorphic as left R-modules.(b) There are infinitely
If p is a prime, let R be the subring The ideal where In is the ideal of Zpn generated by p ϵ Zpn, is a nil ideal of R that is not nilpotent. Σ z of I Zpn. του ηΣ1 n>1
The nil radical N(R) of R is the ideal generated by the set of all nil ideals of R.(a) N(R) is a nil ideal.(b) N(N(R)) = N(R).(c) N(R/ N(R)) = 0.(d) P(R) ⊂ N(R) ⊂ J(R).(e) If R is left Artinian,
A left Artinian ring R has the same number of nonisomorphic simple left R-modules as nonisomorphic simple right R-modules.
Let R be a ring without identity. Embed Rina ring S with identity which has characteristic zero. Prove that J(R) = J(S). Consequently every semi simple ring may be embedded in a semi simple ring with
J(MatnR} = MatnJ(R). Here is an outline of a proof:(a) If A is a left R-module, consider the elements of An = A ⊕ A ⊕· · · ⊕ A (n summands) as column vectors; then An is a left
(a) Let I be a nonzero ideal of R[x] and p(x) a nonzero polynomial of least degree in I with leading coefficient a. If ∫(x) ϵ R[x] and am∫(x) = 0, then am-1p(x}f(x} = 0.(b) If a ring R has no
Let L be a left ideal and Ka right ideal of R. Let M(R) be the ideal generated by all nilpotent ideals of R.(a) L + LR is an ideal such that (L + LR)n ⊂ Ln+ LnR for all n ≥ 1.(b) K + RK is an
If T : ℓ → S is a covariant functor that has a left adjoint, then T is representable.
A morphism in the category of sets is monic [resp. epic] if and only if it is injective [resp. surjective].
Let ℓ be a concrete category and T: ℓ → S the forgetful functor. If T has a left adjoint F : S → ℓ, then F is called a free-object functor and F(X) (X ϵ S) is called a free F-object on
(a) If S : ℓ → D is a functor, let σ{S) = 1 if S is covariant and -1 if S is contravariant. If T: D → ℓ is another functor, show that TS is a functor from ℓ to ℇ whose variance is
Let X be a fixed set and define a functor S : S → S by Y| → X × Y. Then S is a left adjoint of the covariant horn functor hx = homs(X, -).
Show that every pair of functions ƒ, g ; C → D has 'a difference cokernel in the category of sets.
If each square in the following diagram is a pullback- and B' → B is a monomorphism, then the outer rectangle is a pullback. P- A ·e· 1- 'B' B.
Covariant representable functors from S to S preserve surjective maps.
In a category with a zero object, the kernel of a monomorphism is a zero morphism.
(a) K is a perfect field if and only if every field extension of K is separable.(b) (Mac Lane) Assume K is a perfect field, F is not perfect and tr.d.F/ K = 1. Then F is separably generated over K.
Let char K = p ≠ 0 and let u be transcendental over K. Suppose F is generated over K by {u,v1,v2, ... } , where vi is a root of xpi - u ϵ K(u)[x] for i = 1,2, .... Then F is separable over K, but
Proof that If F is algebraically closed and tr.d.F/K is finite, then every K-monomorphism F → F is in fact an automorphism.
Let K = ZP, F = Zp(x), and E = Zp(xP).(a) F is separably generated and separable over K.(b) E ≠ F.(c) F is algebraic and purely inseparable over E.(d) {xP} is a transcendence base of F over K which
If F is algebraically closed and E an intermediate field such that tr.d.E/ K is finite, then any K-monomorphism E → F extends to a K-automorphism of F.
If E is a purely transcendental extension of K and F is algebraic over K, then E and Fare linearly disjoint over K.
(a) If S is a transcendence base of the field C of complex numbers over the field Q of rationals, then S is infinite. (b) There are infinitely many distinct automorphisms of the field C. (c)
If {u1, ... , Un} is algebraically independent over F, then F and K(u1, ... , un) are linearly disjoint over K.
Proof that If F = K(u1, . .. , un) is a finitely generated extension of K and E is an intermediate field, then E is a finitely generated extension of K.
If char K = p ≠ 0, then(a) K1/Pn is a field for every n ≥ 0.(b) K1/P∞ is a field.(c) K1/Pn is a splitting field over K of {xPn - k | k ϵ K}.
Determine the Galois groups of the following polynomials over the fields indicated: (a) x4 - 5 over Q; over (b) (x3 - 2)(x2 - 3)(x2 - 5)(x2 - 7) over Q. (c) x3 - x - 1 over Q; over (d) x3 -
The subring E[F] generated by E and F is a vector space over Kin the obvious way. The tensor product E ⊗K F is also a K-vector space . E and F are linearly disjoint over K if and only if the

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