Let D be a division ring, and let M be a right D-module. Recall that a...

Let D be a division ring, and let M be a right D-module. Recall that a subset S ⊂ M is linearly independent (with respect to D) if for any finite subset T ⊂ S, and elements a_t ∈ D for t ∈ T, if sum of ta_t = 0, then all the a_t = 0.

(a) If S ⊂ M is linearly independent, show that there exists a maximal linearly independent subset U of M that contains S, and that U is a basis for M (that is,M is a free D-module).

(b) Suppose that S is a generating set (that is, for every element m ∈ M, there exists a finite subset T ⊂ S and a_t ∈ D such that m = sum of ta_t). Show that there exists a subset U ⊂ S that is a basis for M.

(c)* Bonus for proving that all bases of M have the same cardinality, if it has a finite basis. (It is also true for infinite bases.)

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Let F be a field, and recall the notion of the characteristic of a ring; the...

Let F be a field, and recall the notion of the characteristic of a ring; the characteristic of a field is either 0 or a prime integer. Show that F has characteristic 0 if and only if it contains a copy of rationals and then F has characteristic p if and only if it contains a copy of the field Z/pZ. Show that (in both cases) this determines the smallest subfield of F.

Suppose that D and E are sets, and D ⊆ E. Let A = P(E). Recall...

Suppose that D and E are sets, and D ⊆ E. Let A = P(E). Recall that P(E) denotes the set of all subsets of E. Define a relation R on A by R = {(X, Y) ∈ A × A: [(X − Y) ∪ (Y − X)] ⊆ D}. So, XRY if and only if [(X−Y) ∪ (Y −X)] ⊆ D. Prove that R is an equivalence relation on A.

In this exercise, we will prove the Division Algorithm for polynomials. Let R[x] be the ring...

In this exercise, we will prove the Division Algorithm for polynomials. Let R[x] be the ring of polynomials with real coefficients. For the purposes of this exercise, extend the definition of degree by deg(0) = −1. The statement to be proved is: Let f(x),g(x) ∈ R[x][x] be polynomials with g(x) ? 0. Then there exist unique polynomials q(x) and r(x) such that f (x) = g(x)q(x) + r(x) and deg(r(x)) < deg(g(x)). Fix general f (x) and g(x). (a) Let...

Let A be a m × n matrix with entries in R. Recall that the row...

Let A be a m × n matrix with entries in R. Recall that the row rank of A means the dimension of the subspace in RN spanned by the rows of A (viewed as vectors in Rn), and the column rank means that of the subspace in Rm spanned by the columns of A (viewed as vectors in Rm). (a) Prove that n = (column rank of A) + dim S, where the set S is the solution space...

Let R be a ring and n ∈ N. Let S = Mn(R) be the ring...

Let R be a ring and n ∈ N. Let S = Mn(R) be the ring of n × n matrices with entries in R. a) i) Let T be the subset of S consisting of the n × n diagonal matrices with entries in R (so that T consists of the matrices in S whose entries off the leading diagonal are zero). Show that T is a subring of S. We denote the ring T by Dn(R). ii). Show...

Let N be a submodule of the R-module M. Prove that there is a bijection between...

Let N be a submodule of the R-module M. Prove that there is a bijection between the submodules of M that contain N and the submodules of M/N.

Let R be a ring and f : M −→ N a morphism of left R-modules. Show that:

Let R be a ring and f : M −→ N a morphism of left R-modules. Show that: c) K := {m ∈ M | f(m) = 0} satisfies the Universal Property of Kernels. d) N/f(M) satisfies the Universal Property of Cokernels. Q2. Show that ZQ :a) contains no minimal Z-submodule

Let R be a ring and f : M −→ N a morphism of left R-modules. Show that:

(3) Let m be a positive integer. (a) Prove that Z/mZ is a commutative ring. (b)...

(3) Let m be a positive integer. (a) Prove that Z/mZ is a commutative ring. (b) Prove that if m is composite, then Z/mZ is not a field. (4) Let m be an odd positive integer. Prove that every integer is congruent modulo m to exactly one element in the set of even integers {0, 2, 4, 6, , . . . , 2m− 2}

2. Let D be a relation on the natural numbers N deﬁned by D = {(m,n)...

2. Let D be a relation on the natural numbers N deﬁned by D = {(m,n) : m | n} (i.e., D(m,n) is true when n is divisible by m. For this problem, you’ll be proving that D is a partial order. This means that you’ll need to prove that it is reﬂexive, anti-symmetric, and transitive. (a) Prove that D is reﬂexive. (Yes, you already did this problem on one of the minihomework assignments. You don’t have to redo the...

Subjects