On uniform dimensions of ideals in right nonsingular rings
A new bimodule invariant is defined, applied to ideals in a right nonsingular ring \(R\), and used to relate the ideal structure of \(R\) to that of its maximal right quotient ring \(Q\). Given a bimodule \(M\), the authors define \(d(M)\) to be the supremum of those positive integers \(n\) such that \(M\) contains an essential right submodule which is a direct sum of \(n\) nonzero sub-bimodules. For example, if \(R\) is semiprime and \(M\) is an ideal in \(R\), then \(d(M)\) equals the uniform dimension of \(_RM_R\). This equality is also verified in a number of other cases; for instance, it holds for ideals in the family \({\mathcal F}(R)\) defined below in case the symmetric maximal quotient ring of \(R\) is semiprime. The invariant \(d\) is particularly well suited to the study of those ideals \(I\) of \(R\) for which there exists an ideal \(J\) with \(I\oplus J\) essential in \(R_R\); this family of ideals is denoted \({\mathcal F}(R)\). Various characterizations of \({\mathcal F}(R)\) are given, and the subfamily of right essentially closed ideals in \({\mathcal F}(R)\) is shown to form a complete Boolean algebra isomorphic to the Boolean algebra of central idempotents in \(Q\). One characterization of \({\mathcal F}(R)\) is that an ideal \(I\) lies in \({\mathcal F}(R)\) if and only if the injective hull \(E(I)\) is an ideal of \(Q\); in this case, it is known that \(E(I)=eQ\) for some central idempotent \(e\in Q\), and so \(E(I)\) is a unital ring. For \(I\in{\mathcal F}(R)\), the authors prove that the ring \(E(I)\) is a direct product of prime rings if and only if there exist ideals \(I_j\) in \(R\), with \(d(I_j)=1\) for all \(j\), such that \(\bigoplus_j I_j\) is an essential submodule of \(I_R\). There is an improved criterion for the case of finite direct products: \(E(I)\) is a direct product of \(n\) prime rings if and only if \(d(I)=n\).
scientific article; zbMATH DE number 1308267
scientific article