×

Fully bounded Noetherian rings of finite injective dimension. (English) Zbl 0691.16037

A fully bounded noetherian ring R is known to be homologically relatively well-behaved. For example, the right global dimension of R equals the supremum of the projective dimensions of the simple right R-modules [J. Rainwater, Commun. Algebra 15, 2143-2156 (1987; Zbl 0628.16010)]. Further, suppose that (1) R contains an uncountable set of central units, the difference of any two of which is also a unit. Then the right Krull dimension of R is bounded above by the right global dimension, provided the latter is finite [the author and R. B. Warfield jun., Proc. Am. Math. Soc. 92, 169-174 (1984; Zbl 0557.16009)].
The aim of the present paper is to prove analogous results involving the injective dimension of R. The author’s methods require (1) as well as (2) R is flat as a module over some central subring, but it is not known whether these hypotheses are necessary. Under these assumptions, he proves that \(inj.\dim (R_ R)\) equals the supremum of those indices i for which \(Ext^ i((R/M)_ R,R_ R)\neq 0\) for some maximal ideal M, and that \(r.K.\dim (R)\leq inj.\dim (R_ R)\) provided the latter is finite. Moreover, if \(inj.\dim (R_ R)=n<\infty\), if P is a prime ideal of R with \(K.\dim (R/P)=k\), and if \(E_ P\) is the injective hull of a uniform right ideal of R/P, then \(E_ P\) does not occur as a direct summand of the ith term of a minimal injective resolution of \(R_ R\) for any \(i>n-k.\)
In case R is an affine noetherian P.I. algebra over a field, or a noetherian P.I. ring satisfying (2), the three results of the previous paragraph also hold.
Reviewer: K.R.Goodearl

MSC:

16E10 Homological dimension in associative algebras
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16P40 Noetherian rings and modules (associative rings and algebras)
16Rxx Rings with polynomial identity
PDFBibTeX XMLCite
Full Text: DOI