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We provide several characterizations and investigate properties of Prüfer modules. In fact, we study the connections of such modules with their endomorphism rings. We also prove that for any Prüfer module M, the forcing linearity number of M, fln(M), belongs to {0,1}.

We introduce the notions of T-Rickart and strongly T-Rickart modules. We provide several characterizations and investigate properties of each of these concepts. It is shown that R is right Σ-t-extending if and only if every R-module is T-Rickart. Also, every free R-module is T-Rickart if and only if $R=Z₂\left(R_R\right)\oplus R^{\text{'}}$, where R’ is a hereditary right R-module. Examples illustrating the results are presented.

Let $I$ be a strong co-ideal of a commutative semiring $R$ with identity. Let ${\Gamma }_I\left(R\right)$ be a graph with the set of vertices $S_I\left(R\right)=\{x\in R\setminus I:x+y\in I$ for some $y\in R\setminus I\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in I$. We look at the diameter and girth of this graph. Also we discuss when ${\Gamma }_I\left(R\right)$ is bipartite. Moreover, studies are done on the planarity, clique, and chromatic number of this graph. Examples illustrating the results are presented.

In this paper, a new kind of graph on a commutative ring is introduced and investigated. Small intersection graph of a ring $R$, denoted by $G\left(R\right)$, is a graph with all non-small proper ideals of $R$ as vertices and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J$ is not small in $R$. In this article, some interrelation between the graph theoretic properties of this graph and some algebraic properties of rings are studied. We investigated the basic properties of the small intersection graph as diameter,...