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Global dimensions of subidealizer rings. (English) Zbl 0833.16004

Let \(R\) be a ring with unity. Many recent papers including J. J. Simón Pinero [Publ. Mat., Barc. 36, No. 1, 189-195 (1992; Zbl 0773.16003)] have studied the computation of \(\text{r.gl.}\dim(R)\) by computing the projective dimension of a relatively few cyclic modules. For any ordinal \(\alpha\), the ring \(R\) is called \(\alpha\)-proper if \(\text{r.K.}\dim(R)\geq\alpha\) and \(\text{r.gl.}\dim(R)=\sup\{\text{pd}(C)\mid C_R\) is \(\beta\)-critical, \(\beta<\alpha\}\), where the terminology on the Krull dimension comes from R. Gordon and J. C. Robson [Mem. Am. Math. Soc. 133 (1973; Zbl 0269.16017)]. If \(S\) is the idealizer of a semimaximal right ideal \(M\) of a ring \(T\) such that \(\text{r.K.}\dim(T)\geq\alpha\geq 1\) and \(T\) is \(\alpha\)-proper, then \(S\) is \(\alpha\)-proper. If \(R\) is a tame subidealizer of a generative right ideal \(M\) of a right noetherian ring \(T\) such that \(\text{r.K.}\dim(R)\geq\alpha\geq 1\), \(\text{K.}\dim(T/M)_R<\alpha\), and \(T\) is \(\alpha\)-proper, then \(R\) is \(\alpha\)-proper. If \(R\) is a tame subidealizer of a generative right ideal \(M\) of a ring \(T\) such that \(\text{r.K.}\dim(T)\geq\alpha\geq 1\), \((T/M)_R\) has finite length, \(\text{Tor}^R_1(A,T/R)=0\) for all \(\alpha\)-critical right modules \(A\), and \(T\) is \(\alpha\)-proper, then \(R\) is \(\alpha\)-proper. In case \(\alpha=1\), these results tell about when idealizers can be computed as \(\text{sup}\{\text{pd}(C_R)\mid C_R\) simple}. Analogous results are also given for the weak global dimension.

MSC:

16E10 Homological dimension in associative algebras
16P40 Noetherian rings and modules (associative rings and algebras)
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
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References:

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