Inverse polynomials and injective covers. (English) Zbl 0794.16004

Park, Sangwon

Commun. Algebra 21, No. 12, 4599-4613 (1993).

Injective covers for \(R\)-modules were defined by Enochs in 1981 using the dual of a categorical definition of an injective envelope. Let \(R\) be a left noetherian ring and \(M\) a left \(R\)-module. Then \(M[x^{-1}]\) may be considered as a left \(R[x]\)-module and it may be proved that if \(M\subset E\) is an injective envelope, then \(M[x^{-1}]\subset E[x^{-1}]\) is an injective envelope over \(R[x]\). The question which motivated the present paper is: If \(E\to M\) is an injective cover, is \(E[x^{-1}]\to M[x^{- 1}]\) an injective cover? The author shows that if \(\overline{E}\to M[x^{-1}]\) is an injective cover, then \(\overline{E}\) has a submodule isomorphic to \(E[x^{-1}]\), providing strong evidence that the answer to the question is in the affirmative.

Reviewer: F.Théron (Pretoria)

MSC:

16D50	Injective modules, self-injective associative rings
16S36	Ordinary and skew polynomial rings and semigroup rings
16P40	Noetherian rings and modules (associative rings and algebras)

Keywords:

inverse polynomials; injective envelope; left noetherian ring; injective cover

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