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The homological grade of a module

Published online by Cambridge University Press:  26 February 2010

David Kirby  and
Hefzi A. Mehran
David Kirby
Affiliation:
Faculty of Mathematical Studies, The University, Southampton, SO9 5NH.
Hefzi A. Mehran
Affiliation:
Department of Mathematics, Kuwait University, Kuwait.
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Extract

For an ideal A of a commutative ring R with identity and a unitary R-module E the notion of an E-sequence of length d in A can be extended as follows. For d = 0 the E-sequence is empty, and for d = 1 it is a subset {ai|i ∈ I} = α ⊆ A such that . For d > 1 we may define, inductively, an E-sequence of length d in A as a sequence

of subsets of A such that a, is an E-sequence of length 1 in A and α2,…, αd is a -sequence of length d − 1 in A. Thus in the standard notion of E-sequence the sets αj, are singletons, and, in effect, the extended notion due to Hochster [1] and Northcott [3] the sets a; are finite. Many of the standard results concerning E-sequences when E is Noetherian extend to the above generalization when the Noetherian condition is dropped. For example it follows from the results of the present note that every maximal E-sequence has the same length (which may be infinite) and every E-sequence can be extended to a maximal E-sequence. This maximal length is inf which we call the homological grade of E in A and denote by hgrR (A; E). So 0 ≤ hgrR (A; E) ≤ ∞, hgrR (A; E) = 0, if, and only if, 0:EA≠0 and hgrg (A; E) = ∞, if, and only if, for all nєℤ.

Type
Research Article
Information
Mathematika , Volume 35 , Issue 1 , June 1988 , pp. 114 - 125
Copyright
Copyright © University College London 1988

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References

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