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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Annihilators of modules with a finite free resolution


Author: Wolmer V. Vasconcelos
Journal: Proc. Amer. Math. Soc. 29 (1971), 440-442
MSC: Primary 13.40
MathSciNet review: 0280478
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Abstract: Let A be a commutative ring and let E be an A-module with a finite free resolution (see below for definition). Extending results known previously for noetherian rings, it is shown that $ {\text{ann}}(E)=\text{annihilator}$ of E is trivial if and only if the Euler characteristic of $ E = \chi (E) > 0$; in addition, if $ \chi (E) = 0,{\text{ann}}(E)$ is dense (i.e. $ {\text{ann}}({\text{ann}}(E)) = 0$). Also, a local ring is constructed with its maximal ideal with a finite free resolution but consisting exclusively of zero-divisors and thus, contrary to the noetherian case, one does not always have a nonzero divisor in $ {\text{ann}}(E)$ if $ \chi (E) = 0$. Finally, if E has a finite resolution by (f.g.) projective modules it turns out that $ {\text{ann}}({\text{ann}}(E))$ is generated by an idempotent element.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0280478-9
PII: S 0002-9939(1971)0280478-9
Keywords: Annihilator, finite free resolution, Euler characteristic, Fitting invariant
Article copyright: © Copyright 1971 American Mathematical Society