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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.
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