Injective hulls with distinct ring structures.
If \(R\) is a ring with \(E=E_R=E(R_R)\) an injective hull and \(\iota\colon R_R\to E_R\) an essential embedding, then the ring structure on \(E(R_R)\) is said to be compatible if the ring multiplication extends the scalar multiplication. It is well known that if \(E(R_R)\) is a rational extension of \(R_R\), then \(E(R_R)\) has a unique compatible ring structure, but \textit{B. L. Osofsky} [Can. Math. Bull. 7, 405-413 (1964; Zbl 0126.06401)] constructed a ring such that no injective hull can possibly have a compatible ring structure. In this paper the question is asked, and answered, whether the existence of a compatible ring structure necessarily implies the uniqueness thereof. A class of right Osofsky compatible rings \(R\) is constructed such that the injective hulls of the rings have more than one ring structure. All of these are isomorphic though, and have the property that they are left Osofsky compatible. The rings used here have \(Q(R)=R\), which is as far as possible from the case where \(Q(R)=E(R_R)\). A final example illustrates the Osofsky compatible ring structures on \(E_R\) using matrix operations.
scientific article; zbMATH DE number 5527494
scientific article