MODULES WHOSE CLOSED SUBMODULES WITH ESSENTIAL SOCLE ARE DIRECT SUMMANDS

Modules whose direct summands are FI-extending

‎A module $M$ is called FI-extending if every fully invariant submodule of $M$ is essential in a direct summand of $M$‎. ‎It is not known whether a direct summand of an FI-extending module is also FI-extending‎. ‎In this study‎, ‎it is given some answers to the question that under what conditions a direct summand of an FI-extending module is an FI-extending module?

Modules Whose Small Submodules Have Krull Dimension

The main aim of this paper is to show that an AB5 module whose small submodules have Krull dimension has a radical having Krull dimension. The proof uses the notion of dual Goldie dimension.

Modules whose hereditary pretorsion classes are closed under products

A module M is called product closed if every hereditary pretorsion class in σ[M ] is closed under products in σ[M ]. Every module which is locally of finite length is product closed and every product closed module is semilocal. LetM ∈ R-Mod be product closed and projective in σ[M ]. It is shown that (1) M is semiartinian; (2) if M is finitely generated then M satisfies the DCC on fully invarian...

Direct Summands of Syzygy Modules of the Residue Class Field

Let R be a commutative Noetherian local ring. This paper deals with the problem asking whether R is Gorenstein if the nth syzygy module of the residue class field of R has a non-trivial direct summand of finite Gdimension for some n. It is proved that if n is at most two then it is true, and moreover, the structure of the ring R is determined essentially uniquely.

Submodules of Secondary Modules