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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

Consider a nitely generatedmodule M overa principal ideal domain R. Let x1, . . . , xn be

...as a finite directsumof indecomposable submodules, so I guess this reduces my question to: must an indecomposable submodule of $M$ be a

You can use Sage to compute with finitelygeneratedmodules (FGM’s) overa principal ideal domain R presented as a quotient V/W, where V and W are free. NOTE: Currently this is only enabled over R=ZZ, since it has not been tested and debugged over more general PIDs. All algorithms make sense...

In this paper, we are interested in classifying nitely generatedmodulesovera principal ideal

2 Finitely-generatedmodulesover noetherian rings. Let R be a commutative ring. Although our immediate interest is in principal ideal domains

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

A finitelygeneratedmoduleovera principal ideal domain is torsion-free if and only if it is free.

4. Smith Normal Form and FinitelyGeneratedModulesoveraPID Before we look at finding limits and colimits of modules, we need to develop some computational tools (the Smith normal form of a matrix) and know how to extract information about modules from homomorphisms between them.

We classify all (finitelygenerated or not) projective modulesovera class of semilocal ring constructed using nealy simple uniserial domains.

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

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...

Suppose that R is aPID and M is a nitely generated R-module. The main example I talked about was R = Z in

Lemma: If is a finitelygenerated torsion moduleovera principal ideal domain, then there exists a nonzero element such that . Proof: By FTFGMPID, we have for some primes and nonnegative natural numbers . Now is nonzero and clearly annihilates . By Proposition 36 on page 396 of D&F...

Any proper submodule of a finitelygeneratedmodule is contained in a maximal submodule.

In this paper we study (non-commutative) rings R over which every nitely generated 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. It is shown that a Noetherian local left FGC-ring is...

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. It is shown that a Noetherian local left...

are finite free modules of the same rank. over. , is an isomorphism. Remark 4. With the notation of Lemma 3

Every finitelygeneratedmodule M overa principal ideal domain R is isomorphic to a unique one of the form.

is a directsumoffinitely many Euclidean rings, we show that this action is transitive if n>k.

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

A module, speaking loosely, is a vector space overa ring instead of overa eld. This statement is justied by examining the dening axioms of a module (in this case we dene

A finitelygeneratedmoduleovera field is simply a finite-dimensional vector space, and a

However, for algebras overaPID, if the algebra is finitelygenerated then so is every subalgebra. Consequently, given a finitelygenerated algebra [math]A[/math], it is a quotient of a free finitelygenerated algebra by some submodule, and that submodule is also finitelygenerated, which gives...

Beginning of classification offinitelygeneratedmodulesoveraPID. Associated Reading: D-F: Chapter 12.

Certainly the classification offinitelygeneratedmodulesoveraPID can be proved in an excessively abstract style. Even so, I considered a great revelation to learn that Jordan canonical form and canonical form for finite abelian groups are really the same theorem. For that matter, I was even more...

Every finitelygeneratedmodule M overa principal ideal domain R is isomorphic to one of the form ⨁ i R / ( q i ) where ( q i ) ≠ R and the ( q i ) are primary ideals. The q i are unique (up to multiplication by units). The elements q i are called the elementary divisors of M. In aPID, nonzero primary ideals are...

Let be a finitelygenerated -module. When is considered as a moduleovera subalgebra of for a subgroup of the group , we write . In Section 2, we show that if is a cyclic -group and the characteristic of does not divide the order of , then we can have a complete system of indecomposable pairwise...

In this paper, we give an answer to the following question of Kaplansky [14] in the local case: For which duo rings R is it true that every finitelygenerated

For every finitelygeneratedmodule M overa principal ideal domain R, there is a unique decreasing sequence of proper ideals. ( d 1 ) ⊇ ( d 2 ) ⊇

Oct 23. Modules. Directsums, direct products.

Algebra is the study of operations, rules and procedures to solve equations. The origin of the term