$\displaystyle δ$-Primary Elements In Lattice Modules

On Generalization of $\displaystyleδ$-Primary Elements in Multiplicative Lattices

In this paper, we introduce $ϕ$-$δ$-primary elements in a compactly generated multiplicative lattice $L$ and obtain its characterizations. We prove many of its properties and investigate the relations between these structures. By a counter example, it is shown that a $ϕ$-$δ$-primary element of $L$ need not be $δ$-primary and found conditions under which a $ϕ$-$δ$-primary element of $L$ is $δ$-primary.

借助若干类型模刻画弱伪准素2-吸收子模

Some Characterizations of Weakly Pseudo Primary 2-Absorbing Submodules in Terms of some Types of Modules

本文研究弱伪准素2-吸收子模，利用忠实、非奇异、Z-正则及投射模刻画乘法模，并在附加条件下证明其子模剩余为相应理想。

I-primary submodules

In this paper, we give a generalization for weakly primary submodules called $I$-primary submodule and we study some properties of it. We give some characterizations of $I$-primary submodules. Also we establish the situation of $I$-primary submodules in module localizations and decomposition modules.

Pseudo-Primary, Classical Prime and Pseudo-Classical Primary Elements in Lattice Modules

In this paper, we introduce the notion of pseudo-primary elements and pseudo-classical primary elements in an $L$-module $M$ and obtain their characterizations. The aim of the paper is to show $rad(N)\in M$, the radical of $N\in M$ is prime if $N$ is either pseudo-primary or pseudo-classical primary. Also, we study classical prime elements of an $L$-module $M$ to obtain many of its characterizations and its properties.

Uniqueness of primary decompositions in Laskerian le-modules

Here we introduce and characterize a new class of le-modules $_{R}M$ where $R$ is a commutative ring with $1$ and $(M,+,\leqslant,e)$ is a lattice ordered semigroup with the greatest element $e$. Several notions are defined and uniqueness theorems for primary decompositions of a submodule element in a Laskerian le-module are established.

On Generalizations of Graded $2$-absorbing and Graded $2$-absorbing primary submodules

Let $R$ be a graded commutative ring with non-zero unity $1$ and $M$ be a graded unitary $R$-module. In this article, we introduce the concepts of graded $ϕ$-$2$-absorbing and graded $ϕ$-$2$-absorbing primary submodules as generalizations of the concepts of graded $2$-absorbing and graded $2$-absorbing primary submodules. Let $GS(M)$ be the set of all graded $R$-submodules of $M$ and $ϕ: GS(M)\rightarrow GS(M)\bigcup\{\emptyset\}$ be a function. A proper graded $R$-submodule $K$ of $M$ is said to be a graded $ϕ$-$2$-absorbing $R$-submodule of $M$ if whenever $x, y$ are homogeneous elements of …