Yücel Tiraş

Prime Modules and Submodules

Abstract In the first section of this paper the authors give some classes of modules where the two concepts π and prime are equivalent. Also, they provide conditions under which a given ring R is a Dedekind domain if and only if the R-module M is prime. In the final section they answer a question concerning modules which satisfy the radical formula.

On Dedekind Modules

In this article the authors give the relation between a finitely-generated torsionfree Dedekind module M over a domain R and prime submodules of the 𝒪(M)-module M and the ring 𝒪(M). They also prove that M is a finitely-generated torsionfree Dedekind module over a domain R if and only if every semi-maximal submodule of R-module M is invertible.

On Invertible and Dense Submodules

Abstract In this paper, the authors give a partial characterization of invertible, dense and projective submodules. In the final section, they give the equivalent conditions to be invertible, dense and projective submodules for a given an R-module M. They also provide conditions under which a given ring R is a Dedekind domain if and only if every non zero submodule of an R-module is locally free.

On prime submodules and primary decomposition

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.

LINEAR GROWTH OF PRIMARY DECOMPOSITIONS OF MODULES AND INTEGRAL CLOSURES

Let be a commutative Noetherian ring with identity. I.Swanson proved in[10] that every ideal in every commutative Noetherian ring has linear growth of primary decompositions. Later, in[8], R.Y. Sharp generalized this result to finitely generated modules over . In the first section we present another (may be simpler) proof of this generalized result. Sharp also proved in[7] that every proper ideal in has linear growth of primary decompositions for integral closures of ideals. We extend, in the last section, this result to finitely generated modules over .