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A simple proof of uniqueness for torsion modules over principal ideal domains.

J. L. García Roig (1985)


The aim of this note is to give an alternative proof of uniqueness for the decomposition of a finitely generated torsion module over a P.I.D. (= principal ideal domain) as a direct sum of indecomposable submodules.Our proof tries to mimic as far as we can the standard procedures used when dealing with vector spaces.For the sake of completeness we also include a proof of the existence theorem.

On commutative rings whose prime ideals are direct sums of cyclics

M. Behboodi, A. Moradzadeh-Dehkordi (2012)

Archivum Mathematicum

In this paper we study commutative rings R whose prime ideals are direct sums of cyclic modules. In the case R is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring ( R , ) , the following statements are equivalent: (1) Every prime ideal of R is a direct sum of cyclic R -modules; (2) = λ Λ R w λ where Λ is an index set and R / Ann ( w λ ) is a principal ideal ring for each λ Λ ; (3) Every prime ideal of R is a direct sum of at most...

On prime submodules and primary decomposition

Yücel Tiraş, Harmanci, Abdullah (2000)

Czechoslovak Mathematical Journal

We characterize prime submodules of R × R for a principal ideal domain R and investigate the primary decomposition of any submodule into primary submodules of R × R .

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