David E. Rush

Strongly irreducible ideals of a commutative ring

Abstract An ideal I of a ring R is said to be strongly irreducible if for ideals J and K of R, the inclusion J∩K⊆I implies that either J⊆I or K⊆I. The relationship among the families of irreducible ideals, strongly irreducible ideals, and prime ideals of a commutative ring R is considered, and a characterization is given of the Noetherian rings which contain a non-prime strongly irreducible ideal.

Reductions of modules

Let △ be a multiplicatively closed set of finitely generated nonzero ideals of a ring R. Then the concept of a △ -reduction of an R -submodule D of an R -module A is introduced and several basic properties of such reductions are established. Among these are that a minimal △ -reduction B of D exists and that every minimal basis of B can be extended to a minimal basis of all R -submodules between B and D, when R is local and A is a finite R -module. Then, as an application, △ -reductions B of a submodule C with property (∗) are introduced, characterized, and shown to be quite plentiful. Here, (∗) means that (R ,M) is a local ring of altitude at least one, that △ = {Mn ; n ≥ 0} and that if D ⊆ E are R -submodules between B and C, then every minimal basis of D can be extended to a minimal basis of E.

Rings with two-generated ideals

Abstract A commutative ring R is said to have the two-generator property if each ideal of R can be generated by two elements, and is said to be stable if each regular ideal of R is projective over its endomorphism ring. It is known that for a one-dimensional local Macaulay ring the two-generator property implies stability but not conversely. We extend some of the known results on rings with the two-generator property to stable rings, and determine conditions under which stable rings have the two-generator property. We also extend a structure theorem for certain finitely generated torsionfree modules over rings with the two-generator property, and some of its consequences for cancellation of direct summands, to one-dimensional rings which may not have finite integral closure, and remove the finite integral closure hypothesis from a characterization of Greither of commutative group rings with the two-generator property.

The Finiteness of I When R[ X ]/I is Flat

Let R be a commutative ring with identity, let X be an indeterminate, and let I be an ideal of the polynomial ring R[X]. Let min I denote the set of elements of I of minimal degree and assume henceforth that min I contains a regular element. Then R[XI/I is a flat R-module implies I is a finitely generated ideal. Under the additional hypothesis that R is quasi-local integrally closed, the stronger conclusion that I is principal holds. (An example shows that the first statement is no longer valid when min I does not contain a regular element.) Let c(I) denote the content ideal of I, i.e. c(I) is the ideal of R generated by the coefficients of the elements of I. A corollary to the above theorem asserts that R[X]/I is a flat R-module if and only if I is an invertible ideal of R[X] and c(I) = R. Moreover, if R is quasi-local integrally closed, then the following are equivalent: (i) R[X1/I is a flat R-module; (ii) R[X]/I is a torsion free R-module and c(I) = R; (iii) I is principal and c(I) = R. Let 6 denote the equivalence class of X in R[XI/I, and let (1, 6 . 4t denote the R-module generated by 1, 6, ***, ft. The following statements are also equivalent: (i) (1, 4, .., t) is flat for all t ? 0; (ii) (1, , ..., t) is flat for some t > 0 for which 1 -, .. t are linearly dependent over R; (iii) I = (f , , fn), fi e min I, and c(I) R; (iv) c(min I) = R. Moreover, if R is integrally closed, these are equivalent to R[X]/I being a flat R-module. A certain symmetry enters in when f is regular in R[L], and in this case (i)-(iv) are also equivalent to the assertion that R[L] and R[1/f are flat R-modules. The main results of this paper are found in ??2 and 3. Many of the difficulties of these sections already occur when the ring R is an integral domain, and the reader might benefit by first confining his attention to this case. Additional technical difficulties arise when one proceeds to the case that R is an arbitrary ring and I is subject to the restriction that min I contain a regular element. ?4 is devoted to a discussion of what happens when one removes this condition on min I. In particular, we make there a conjecture as to the class of rings R with the property that for every ideal I of R[X], if R[X]/I is R-flat, then I is finitely generated. The corresponding question for finitely generated modules is easily answered as follows: The class of rings R with the property that R[X]/I is a finite flat R-module implies I is finitely generated is exactly Presented to the Society, March 27, 1971; received by the editors May 10,1971. AMS 1970 subject classifications. Primary 13A15, 13B25, 13C10; Secondary 13B20.

An arithmetic characterization of algebraic number fields with a given class group

Let R be the ring of integers of a number field K with class group G . It is classical that R is a unique factorization domain if and only if G is the trivial group; and the finite abelian group G is generally considered as a measure of the failure of unique factorization in R . The first arithmetic description of rings of integers with non-trivial class groups was given in 1960 by L. Carlitz (1). He proved that G is a group of order ≤ two if and only if any two factorizations of an element of R into irreducible elements have the same number of factors. In ((6), p. 469, problem 32) W. Narkiewicz asked for an arithmetic characterization of algebraic number fields K with class numbers ≠ 1, 2. This problem was solved for certain class groups with orders ≤ 9 in (2), and for the case that G is cyclic or a product of k copies of a group of prime order in (5). In this note we solve Narkiewiczs problem in general by giving arithmetical characterizations of a ring of integers whose class group G is any given finite abelian group.

Note on Noetherian filtrations

Let R be a commutative ring with a filtration . Then f is said to be Noetherian in case the associated graded ring is Noetherian. The main result gives several characterizations of a Noetherian filtration on a Noetherian ring, and these are used to correct some erroneous results in the literature. They are then extended to filtrations on modules, and then these generalizations are tied in to some related results in the literature.

Note on i-good filtrations and noetherian rees rings

If f = (In )n≥1 is filtration on a commutative ring R and E is an R -module, then a filtration e = (E n )n≥1 on E is called f -good in case I m I n ⊂ Em+n for all m≥1 and n ≥ 1 and there exists a positive integer m such that for all n >m (This is an extension of the usual definition of e being i -good with i an ideal in R) A number of the basic results concerned with R -good filtrations are extended to f -good filtrations, and it is then shown that if f = (In )n≥1 and k = (Kn )n ≥1 are regular filtrations on R such that Kn ⊆ I n for all n ≥ 1, and if k is f -good, then the complete integral closures of the Rees rings of f and k are equal.

Asymptotic Primes of Delta Closures of Ideals

Abstract For an ideal H in a Noetherian ring R let H* = ∪{H i+1 : R H i | i ≥ 0} and for a multiplicatively closed set Δ of nonzero ideals of R let H Δ = ∪{HK: R K | K ∈ Δ}. It is shown that four standard results concerning the associated prime ideals of the integral closure (bR)a of a regular principal ideal bR do not hold for certain Δ closures (bR)Δ of bR. To do this it is first shown that if I is an ideal in R such that height (I) ≥ 1, then each radical ideal J of R containing I is of the form J = K* :R cR for some ideal K closely related to I, and if I a :R J ⊈ U = ∪{I*R P ∩ R | P is a minimal prime divisor of J} (where I a is the integral closure of I), then J = I Δ :R CR and I ⊆ I Δ ⊆ I a).

Bezout domains with stable range 1

Abstract It is shown that certain classes of Bezout domains have stable range 1, and thus are elementary divisor rings. Included is a strengthening of Roquettes principal ideal theorem which states that the holomorphy ring of a family S of valuation rings of a field K, with S having bounded residue fields, is Bezout. A counterpart is also given where a bound is placed on the ramification indices instead of the residue fields, and these results are applied to rings of integer-valued rational functions over these rings. Along the way, characterizations are given of Prufer domains with torsion class group, Bezout domains, and Bezout domains with stable range 1 in terms of a family { B (t) | t∈K} of numerical semigroups associated with the ring R, and a related family { D (t) | t∈K} of numerical semigroups.

Noetherian properties in monoid rings

Abstract Let R be a commutative unitary ring and let M be a commutative monoid. The monoid ring R [ M ] is considered as an M -graded ring where the homogeneous elements of degree s are the elements of the form aX s , a ∈ R , s ∈ M . If each homogeneous ideal of R [ M ] is finitely generated, we say R [ M ] is gr-Noetherian. We denote the set of homogeneous prime ideals of R [ M ] by h-Spec( R [ M ]). Results are given which illuminate the difference between the Noetherian and gr-Noetherian conditions on a monoid ring, and also the difference between Spec( R [ M ]) being Noetherian and h-Spec( R [ M ]) being Noetherian. Applications include a variation of the Mori–Nagata theorem and some results on group rings which are ZD-rings, Laskerian rings or N-rings.