On Continuous Hulls of Rings and Modules

Abstract

Every module has an injective hull, a quasi-injective hull, and a quasi-continuous hull. However, there exists a uniform nonsingular cyclic module over a noncommutative ring which has no continuous hull. (Recall that the continuous hull of a module M M is the smallest continuous extension of M M in a fixed injective hull E ( M ) E(M) of M M .) When the base ring is commutative, every uniform cyclic module over a commutative ring has a continuous hull. Further, every nonsingular cyclic module over a commutative ring has a continuous hull. Motivated by the example and these results on continuous hulls, it is interesting to study whether R R R_R has a continuous hull for any ring R R . We show that for any integer n n such that n > 1 n>1 , if R R is the n × n n\times n matrix ring over any given ring or the n × n n\times n upper triangular matrix ring over any given ring, then R R R_R has a continuous hull and such a continuous hull is explicitly described. Moreover, if R R is an abelian regular ring, it is shown that every nonsingular cyclic right R R -module has a continuous hull. As an application, we show that R R R_R has a continuous hull when R R is an abelian regular ring. In this case, R B ( Q ( R ) ) R R\mathcal {B}(Q(R))_R (where R B ( Q ( R ) ) R\mathcal {B}(Q(R)) is the idempotent closure of R R ) is the continuous hull of R R R_R . As a byproduct, we show that when R R is an abelian regular ring, R B ( Q ( R ) ) R\mathcal {B}(Q(R)) is the smallest continuous regular ring of quotients of R R . We discuss the condition ( ⋆ \star ) of a commutative ring R R , which is precisely that the classical ring of quotients of every uniform factor ring of R R is self-injective. Moreover, by using the condition ( ⋆ \star ), we provide a detailed proof that every module over a commutative noetherian ring has a continuous hull. Various examples which illustrate our results are provided.