The pure-injective and RD-injective hulls of a ring
\textit{A. Facchini} proved [Q. J. Math., Oxf. II. Ser. 39, 307-321 (1988; Zbl 0668.13012)] that if \(R\) is a commutative ring and if the pure- injective hull of \(R\) (as a module over itself) is indecomposable then \(R\) is local and he asked if the converse is true. The author shows that, in fact, if \(R\) is any ring then its pure-injective hull as a (right, say) module over itself is indecomposable if and only if \(R\) is local. Indeed, \textit{I. Herzog} has shown more generally [Contemp. Math. 130, 153-165 (1992; Zbl 0797.16019)] that if \(M\) is a finitely presented module (over any ring) then the pure-injective hull of \(M\) is indecomposable if and only if \(M\) has local endomorphism ring. An embedding of the module \(M\) into the module \(N\) is said to be RD-pure if, for every element \(r\) of the ring, \(Mr=M \cap Nr\). There is a corresponding notion of RD-injective and [see \textit{A. Facchini}, J. Algebra 110, 380-406 (1987; Zbl 0629.13008)] RD-injective hull. The author provides a criterion for the (right) RD-injective hull of a ring to be indecomposable. He uses this to show that if \(R\) has indecomposable (left or right) RD-injective hull then \(R\) is local and he shows that the converse is false by giving an example of a (local artinian) ring whose right RD-injective hull is indecomposable but whose left RD-injective hull is not. He also notes that the ring \(k[x,y : x^ 2=y^ 2=xy=0]\), where \(k\) is a field, is pure-injective but not RD- injective as a module over itself.
scientific article; zbMATH DE number 578713