Let [Formula: see text] be a commutative ring and [Formula: see text] an [Formula: see text]-module. Let [Formula: see text] and [Formula: see text]. [Formula: see text] satisfies Property [Formula: see text] (respectively, Property [Formula: see text]) if for each finitely generated ideal [Formula: see text] (respectively, finitely generated submodule [Formula: see text]) ann[Formula: see text] (respectively, ann[Formula: see text]). The ring [Formula: see text] satisfies Property [Formula: see text] if [Formula: see text] does. We study rings and modules satisfying Property [Formula: see text] or Property [Formula: see text]. A number of examples are given, many using the method of idealization.
Let R be a commutative ring with identity and let P (R) be the monoid of principal fractional ideals of R. We show that P (R) is finitely generated if and only if P (R) (R the integral closure of R) is finitely generated andR/[R :R] is finite. Moreover,R is a finite direct product of finite local rings, SPIRs, Bezout domains D with P (D) finitely generated, and special Bezout rings S (S is a Bezout ring with a unique minimal prime P , S P is an SPIR, and P (S/P ) is finitely generated). Also, P (R) is finitely generated if and only if F * (R), the monoid of finitely generated fractional ideals of R, is finitely generated. We show that the monoid F (R) of fractional ideals of R is finitely generated if and only if the monoidF (R) of R-submodules of the total quotient ring of R is finitely generated and characterize the rings for which this is the case.
In this paper we investigate principal prime ideals in commutative rings. Among other things we characterize the principal prime ideals that are both minimal and maximal and characterize the maximal ideals of a polynomial ring that are principal. Our main result is that if ( p) is a principal prime ideal of an atomic ring R, then ht( p) ≤ 1.2000 Mathematics subject classification: primary 13A15.
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