## Bulletin of the Belgian Mathematical Society - Simon Stevin

### A Generalization of Rickart Modules

#### Abstract

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 Rickart modules. $\pi$-Rickart modules are also a dual notion of dual $\pi$-Rickart modules and extends that of generalized right principally projective rings to the module theoretic setting. The module $M$ is called {\it $\pi$-Rickart} if for any $f\in S$, there exist $e^2=e\in S$ and a positive integer $n$ such that $r_M(f^n)=$ Ker$f^n=eM$. We obtain several results about generalized right principally projective rings by using $\pi$-Rickart modules. Moreover, we investigate relations between a $\pi$-Rickart module and its endomorphism ring.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 21, Number 2 (2014), 303-318.

Dates
First available in Project Euclid: 20 May 2014

https://projecteuclid.org/euclid.bbms/1400592627

Digital Object Identifier
doi:10.36045/bbms/1400592627

Mathematical Reviews number (MathSciNet)
MR3211018

Zentralblatt MATH identifier
1305.16001

#### Citation

Ungor, Burcu; Halıcıoglu, Sait; Harmanci, Abdullah. A Generalization of Rickart Modules. Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 2, 303--318. doi:10.36045/bbms/1400592627. https://projecteuclid.org/euclid.bbms/1400592627