A SUBCLASS OF BAER IDEALS AND ITS APPLICATIONS

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Document Type : Original Manuscript

Authors

Department of Mathematics, Yasouj University, Yasouj, Iran.

Abstract

An ideal $I$ of a ring $R$ is called a right strongly Baer ideal if $r(I)=r(e)$, where $e$ is an idempotent, and there are right semicentral idempotents $e_{i}$ ($1\leq i\leq n$) with $ReR=Re_{1}R\cap Re_{2}R\cap...\cap Re_{n}R$ and each ideal $Re_{i}R$ is maximal or equals $R$. In this paper, we provide a topological characterization of this class of ideals in semiprime (resp., semiprimitive) rings. By using these results, we prove that every ideal of a ring $R$ is a right strongly Baer ideal \textit{if and only if} $R$ is a semisimple ring. Next, we give a characterization of right strongly Baer-ideals in 2-by-2 generalized triangular matrix rings, full and upper triangular matrix rings, and semiprime rings. For a semiprimitive commutative ring $R$, it is shown that $\Soc(R)$ is a right strongly Baer ideal \textit{if and only if} the set of isolated points of $\Max(R)$ is dense in it \textit{if and only if} $\Soc_{m}(R)$ is a right strongly Baer ideal. Finally, we characterize strongly Baer ideals in $C(X)$ (resp., $C(X)_{F}$).

Keywords

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Journal of Algebraic Systems

Articles in Press, Accepted Manuscript
Available Online from 09 April 2024

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