Published online by Cambridge University Press: 26 February 2010
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.
To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.
To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.