Vol. 36, No. 3, 1971

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Completions of Dedekind prime rings as second endomorphism rings

James J. Kuzmanovich

Vol. 36 (1971), No. 3, 721–729

The purpose of this paper is to show that if M is a maximal two-sided ideal of a Dedekind prime ring R and P is any maximal right ideal containing M, then the M-adic completion R of R can be realized as the second endomorphism ring of E = E(R∕P), the R-injective hull of R∕P; that is, as endK(E) where K = end(ER). The ring K turns out to be a complete, local, principal ideal domain.

This paper was motivated by a result of Matlis [6] which says that if P is a prime ideal of a commutative Noetherian ring R, then the P-adic completion of the localization of R at P can be realized as the ring of endomorphisms of E = E(R∕P), the R-injective hull of R∕P.

Since R is a ful] matrix ring over a complete local domain L [4], we are able to approach the problem by considering first the case that R is a complete local domain, then by means of the Morita theorems we pass to the case R = R, and finally pass to the general case.

Mathematical Subject Classification
Primary: 16.53
Received: 8 April 1970
Published: 1 March 1971
James J. Kuzmanovich