When Some Complement of an EC-Submodule is a Direct Summand

Modules for which every non-cosingular submodule is a summand

‎A module $M$ is lifting if and only if $M$ is amply supplemented and‎ ‎every coclosed submodule of $M$ is a direct summand‎. ‎In this paper‎, ‎we are‎ ‎interested in a generalization of lifting modules by removing the condition‎"‎amply supplemented‎" ‎and just focus on modules such that every non-cosingular‎ ‎submodule of them is a summand‎. ‎We call these modules NS‎. ‎We investigate some gen...

SOLVING FUZZY LINEAR SYSTEMS BY USING THE SCHUR COMPLEMENT WHEN COEFFICIENT MATRIX IS AN M-MATRIX

This paper analyzes a linear system of equations when the righthandside is a fuzzy vector and the coefficient matrix is a crisp M-matrix. Thefuzzy linear system (FLS) is converted to the equivalent crisp system withcoefficient matrix of dimension 2n × 2n. However, solving this crisp system isdifficult for large n because of dimensionality problems . It is shown that thisdifficulty may be avoide...

The Complement of a D-Tree is Pure Shellable

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?

Heitmann ’ S Proof of the Direct Summand

We describe the main ideas of Ray Heitmann’s proof of the Direct Summand Conjecture in dimension 3 for a ring of mixed characteristic [1]. In the first section we describe the main methods which are used and prove the necessary lemmas. In the second section we prove the main result of Heitmann’s paper. Finally, in the third section we give a proof of the Canonical Element Conjecture using this ...