Reductions of ideals in commutative rings

Author:
James H. Hays

Journal:
Trans. Amer. Math. Soc. **177** (1973), 51-63

MSC:
Primary 13A15

DOI:
https://doi.org/10.1090/S0002-9947-1973-0323770-8

MathSciNet review:
0323770

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Abstract: All rings considered in this paper are commutative, associative, and have an identity. If *A* and *B* are ideals in a ring, then *B* is a reduction of *A* if and if for some positive integer *n*. An ideal is basic if it has no reductions. These definitions were considered in local rings by Northcott and Rees; this paper considers them in more general rings. Basic ideals in Noetherian rings are characterized to the extent that they are characterized in local rings. It is shown that elements of the principal class generate a basic ideal in a Noetherian ring. Prüfer domains do not have the basic ideal property, that is, there may exist ideals which are not basic; however, a characterization of Prüfer domains can be given in terms of basic ideals. A domain is Prüfer if and only if every finitely generated ideal is basic.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0323770-8

Keywords:
Reductions of ideals,
basic,
Noetherian rings,
Prüfer domains,
valuation,
valuation ring,
*C*-ideals,
analytically independent,
principal class,
Dedekind domains

Article copyright:
© Copyright 1973
American Mathematical Society