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Projective Ideals of Finite Type

Published online by Cambridge University Press:  20 November 2018

William W. Smith
William W. Smith*
Affiliation:
The University of North Carolina, Chapel Hill, North Carolina
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The main results in this paper relate the concepts of flatness and projectiveness for finitely generated ideals in a commutative ring with unity. In this discussion the idea of a multiplicative ideal is used.

Definition.An ideal Jis multiplicative if and only if whenever I is an ideal with I ⊂ J there exists an ideal Csuch that I = JC.

Throughout this paper Rwill denote a commutative ring with unity. If I and Jare ideals of R,then I: J = {x| xJ ⊂ I}. By “prime ideal” we will mean “proper prime ideal” and Specie will denote this set of ideals. Ris called a local ring if it has a unique maximal ideal (the ring need not be Noetherian). If P is in Spec R then RPis the quotient ring formed using the complement of P.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1057 - 1061
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Bourbaki, N., Eléments de mathématique, Fasc. 27, Algèbre commutative] chapitre 1: Modules plats; chapitre 2: Localisation; Actualités Sci. Indust., No. 1290 (Hermann, Paris, 1961).Google Scholar
2. Cartan, H. and Eilenberg, S., Homological algebra (Princeton Univ. Press, Princeton, N.J., 1956).Google Scholar
3. MacRae, R. E., On an application of the Fitting invariants, J. Algebra 2 (1965), 153–169.Google Scholar