A SUBCLASS OF BAER IDEALS AND ITS APPLICATIONS

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} (۱\leq i\leq n) with ReR=Re_{۱}R\cap Re_{۲}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 ۲-by-۲ 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}).

کلیدواژه ها:

Traingular matrix ring ، idempotent element ، socle of a ring ، ring of continuous function ، Zariski topology