Non-self-injective injective hulls with compatible multiplication. (Q2382994)

Let \(R\) be a ring and \(S\) the injective hull of the regular right module \(R\). The module \(S\) can often be made into a ring with multiplication compatible with that of \(R\). This happens, for instance, when \(R\) is right non-singular and then \(S\) is the maximal right ring of quotients of \(R\). Also, if \(S\) is a rational extension of \(R\), then \(S\) has a unique ring structure. \textit{B. Osofsky} [Can. Math. Bull. 7, 405-413 (1964; Zbl 0126.06401)] asked if, in such a situation, the right regular module \(S\) is necessarily injective. The authors construct some interesting examples giving a negative answer to this question and, moreover, construct an infinite chain of such rings.