A note on finite union of primary submodules

A note on finite union of primary submodules

  • Esra Şengelen Sevim
  • Tarık Arabacı
  • Unsal Tekir
  • Suat Koc

Abstract

Let $M\ $be a nonzero unital $R$-module. A proper submodule $N\ $of $M\ $is
said to be a primary submodule if for $a\in R,m\in M\ $and whenever $am\in
N,\ $then either $a\in\sqrt{(N:M)}\ $or $m\in N.\ $Atani and Tekir gave the
Primary Avoidance Theorem as follows: let $N\ $be a submodule of $M$ and
$N\subseteq%
%TCIMACRO{\tbigcup \limits_{i=1}^{n}}%
%BeginExpansion
{\textstyle\bigcup\limits_{i=1}^{n}}
%EndExpansion
N_{i}$ be a covering of submodules of $M,$ where at most two of $N_{i}$'s are
not primary. If $\sqrt{(N_{i}:M)}\nsubseteq\sqrt{(N_{j}:M)}$ for all $i\neq
j,\ $then $N\subseteq N_{k}\ $for some $k\in\{1,2,\ldots,n\}$ \cite[Theorem
1]{AtTe}$.\ $In this paper, our aim is to improve the aforementioned version of Primary Avoidance Theorem and to obtain a similar result with weaker conditions.

Published
2019-04-30
How to Cite
ŞENGELEN SEVIM, Esra et al. A note on finite union of primary submodules. Eurasian Bulletin of Mathematics, [S.l.], v. 2, n. 1, p. 32-35, apr. 2019. Available at: <http://www.ebmmath.com/index.php/EBM/article/view/38>. Date accessed: 23 sep. 2019.