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In short, the proposition says that every finitelygeneratedmoduleoveraPID is comprised of a part which is free (i.e. has a basis) and a

Lemma 1 If is a pure cyclic submodule of a finitelygeneratedmodule and is a directsumof cyclic modules then . Throughout the lecture today we assume that is aPID, and is a finitelygeneratedmoduleover . a finite directsumof indecomposable submodules, so I guess this reduces my question to: must an indecomposable submodule of $M$ be a

2 Finitely-generatedmodulesover noetherian rings. Let R be a commutative ring.

Now K is a submodule of a Noetherian module; hence K is nitely generated. Pick a nite set of generators of K (it turns

Finite dimensional vector fields over are all of the form , The classification theorem for finitelygenerated abelian groups

1. Denition of modules, homomorphisms, submodules, quotient modules, directsums and products of modules.

Finitelygeneratedmodule V/W over Integer Ring with invariants (4, 16). The invariants are computed using the Smith normal form algorithm, and determine the

In this section, and the following section, we will determine the structure of nitely generatedmodulesovera principal ideal domain. We begin in this section by considering nitely generated free mod-ules. The following simple lemma is valid for modulesover any ring with identity; it should go in an...

2. Finitely-generatedmodulesovera domain. In the sequel, the results will mostly require that R be a

If is aPID, then every finitelygeneratedmoduleover is isomorphic to a directsumof cyclic -modules. That is, there is a unique decreasing sequence of proper ideals such that where , and . Similarly, every graded moduleovera graded PID decomposes uniquely into the form where are...

Let $M^n$ be a finitelygenerated free module of dimension $n > 0$ overa principal ideal domain. I am trying to prove that for every non-zero element $a$ o.

Abstract: In this paper we study (non-commutative) rings $R$ over which every finitelygenerated left module is a directsumof cyclic modules (called left FGC-rings). The commutative case was a well-known problem studied and solved in 1970s by various authors.

is a directsumof a finite free module and a torsion module Mtors. which is a summand of a module of the form ⨁i=1,…,nR/fiR. with f1,…,fn∈R.

The notion of a moduleovera commutative ring is a generalization of the notion of a vector space overa eld, where the scalars belong

Conversely, a directsumofdirectsums corresponding to distinct (non-associate) primes in R can be reassembled in a unique manner to t the

Homomorphisms, quotients, directsums and products.

Modules: finitelygeneratedmodulesoveraPID: canonical forms for matrices: Jordan canonical form.