In this paper, we study on the primeness and semiprimeness of a Morita context related to the rings and modules. Necessary and sufficient conditions are investigated for an ideal of a Morita context to be a prime ideal and a semiprime ideal. In parti
cular, we determine the conditions under which a Morita context is prime and semiprime.
Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S=$ End$_R(M)$. Let $Z_2(M)$ be the second singular submodule of $M$. In this paper, we define Goldie Rickart modules by utilizing the endomorphisms of a module. The module $
M$ is called Goldie Rickart if for any $fin S$, $f^{-1}(Z_2(M))$ is a direct summand of $M$. We provide several characterizations of Goldie Rickart modules and study their properties. Also we present that semisimple rings and right $Sigma$-$t$-extending rings admit some characterizations in terms of Goldie Rickart modules.
Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S=$ End$_R(M)$. In this paper we introduce $pi$-Rickart modules as a generalization of generalized right principally projective rings as well as that of Rickart modules. The
module $M$ is called {it $pi$-Rickart} if for any $fin S$, there exist $e^2=ein S$ and a positive integer $n$ such that $r_M(f^n)=eM$. We prove that several results of Rickart modules can be extended to $pi$-Rickart modules for this general settings, and investigate relations between a $pi$-Rickart module and its endomorphism ring.
