Rings over which each module possesses a maximal submodule
Let \(A\) be an associative ring with nonzero identity element. \(A\) is said to be a right Bass ring if any nonzero right module over \(A\) has a maximal submodule. By \(C(A)\) is denoted the center of the ring \(A\). The ring \(A\) is said to be algebraic over its center if for every element \(a\in A\), there is a polynomial \(f\in C(A)[x]\) such that \(f(a)=0\) and the leading coefficient of the polynomial \(f(x)\) is a regular element of the ring \(A\). The main result established by the author is: Assume that any prime quotient ring of the ring \(A\) is algebraic over its center. Then the following conditions are equivalent: (1) \(A\) is a right perfect ring; (2) \(A\) is a right Bass ring that contains no infinite set of orthogonal idempotents. Also, a series of other results on right Bass rings is proved.
scientific article; zbMATH DE number 1238067