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Indecomposable injective modules over a commutative noetherian ring have a very nice structure, as given by a famous theorem of E. Matlis. In particular, if K is a field and R_n=K[x₀,…,x_n-1], the polynomial ring in n commuting indeterminates, and I is any ideal generated by some of the indeterminates, then the injective envelope of R_n/I has a very nice description in terms of the inverse polynomial construction of Macualay and Northcott.

The ring R_ω =K[x₀,…,x_i,…; i<ω] is coherent but not noetherian; on the surface it appears to be a close relative of the rings R_n. Lorna Gregory, a PhD student of Mike Prest in Manchester, has been looking at the injective envelope of R_ω/M, where M is the maximal ideal generated by all the indeterminates. It is not just a "complexified" version of its noetherian relatives. I have a description of a large and complicated essential extension E_o of R_ω/M, and Lorna has some nice complexity results, showing how far E_o must be from the injective envelope.

REFERENCES:

1. E. Matlis, Injective Modules over Noetherian Rings, Pacific J. Math (8) 511--528, 1958.
2. D. G. Northcott, }, Injective Envelopes and Inverse Polynomials, Journal of the LMS, Series 2, (8) 290--296, 1974.
3. T. G. Kucera, Explicit descriptions of injective envelopes: generalizations of a result of Northcott, Comm. Algebra, 17(11) 2703--2715, 1989.