Moshe Roitman

Polynomial extensions of atomic domains

Abstract Among other results, we obtain here a normal atomic domain A such that A [ X ] is not atomic but any reducible element in A is a product of two irreducible elements (an atomic domain is a domain in which every nonzero nonunit is a product of irreducible elements).

On Mori domains and commutative rings with CC⊥ I

Abstract We prove here, among other results, that if A is a Mori domain containing an uncountable field, then any polynomial ring over A is Mori. The Mori property of a domain is determined by its maximal ideals. We use extensively a known characterization of Mori domains in terms of the chain condition on annihilators. We characterize Mori domains in terms of semivaluations and divisibility groups. We prove an analogue of Nagatas theorem for the Mori property and present an example of a completely integrally closed domain A , S a multiplicative subset of A generated by prime elements, such that A is completely integrally closed, but A S is not.

The content of a Gaussian polynomial is invertible

Let R be an integral domain and let f(X) be a nonzero polynomial in R[X]. The content of f is the ideal c(f) generated by the coefficients of f. The polynomial f(X) is called Gaussian if c(fg) = c(f)c(g) for all g(X) ∈ R[X]. It is well known that if c(f) is an invertible ideal, then f is Gaussian. In this note we prove the converse.

A characterization of cancellation ideals

An ideal I of a commutative ring R with identity is called a cancellation ideal if whenever IB = IC for ideals B and C of R, then B = C. We show that an ideal I is a cancellation ideal if and only if I is locally a regular principal ideal. Let R be a commutative ring with identity. An ideal I of R is called a cancellation ideal if whenever IB = IC for ideals B and C of R, then B = C. It is easily seen that I is a cancellation ideal if and only if whenever IB ⊆ IC for ideals B and C of R, then B ⊆ C. A good introduction to cancellation ideals may be found in Gilmer [1, Section 6]. As for examples, it is easy to see that a principal ideal (a) is a cancellation ideal if and only if (a) is a regular ideal (i.e., a is not a zero divisor). An invertible ideal is a cancellation ideal. More generally, an ideal that is locally a regular principal ideal is a cancellation ideal. The purpose of this paper is to prove the converse. Kaplansky [2, Theorem 287] proved that a finitely generated cancellation ideal in a quasi-local domain is principal. We begin with the following lemma which is a modification of Kaplansky’s result (see [1, Exercise 7, page 67]). We use essentially the same argument. Lemma. Let R be a commutative ring with identity and let I be a cancellation ideal of R. Suppose that I = (x, y)+A where A is an ideal of R containing MI for some maximal ideal M . Then I = (x) +A or I = (y) +A. Proof. Put J = ( x + y, xy, xA, yA,A ) . Then it is easily checked that IJ = I. Since I is a cancellation ideal, we have J = I. Thus x = λ ( x + y ) + terms from ( xy, xA, yA,A ) . First, suppose that λ ∈ M . Since λx ∈ MI ⊆ A, we have x ∈ (y2, xy, xA, yA,A2). LetK = (y)+A. Then I = IK. Since I is a cancellation ideal, we have I = K. Next, suppose that λ / ∈M . Then for some μ ∈ R andm ∈M , we have μ (−λ) = 1+m. Now −μλy2 = μ (λ− 1)x+ terms from (xy, xA, yA,A2). Since my = (my) y ∈ (MI) y ⊆ Ay, we have y ∈ (x2, xy, xA, yA,A2). Thus, as in the first case, we get that I = (x) +A. Theorem. Let R be a commutative ring with identity. An ideal I of R is a cancellation ideal if and only if I is locally a regular principal ideal. Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 13A15.

On polynomial extensions of Mori domains over countable fields

Abstract We show here that a polynomial extension of a Mori domain is not necessarily Mori. More precisely, given a field K , there exists a Mori domain A containing K such that A [ X ] is not Mori if and only if K is countable. A similar statement holds for the CC ⊥ property.

On hilbert functions of reduced and of integral algebras

Abstract We prove that for any finite field k , there exist differentiable O -sequences which are not Hilbert functions of reduced graded k -algebras. We discuss when generic Hilbert functions and the Hilbert function of a complete intersection can be Hilbert functions of reduced or of integral graded algebras.

Finite generation of powers of ideals

Suppose M is a maximal ideal of a commutative integral domain R and that some power Mn of M is finitely generated. We show that M is finitely generated in each of the following cases: (i) M is of height one, (ii) R is integrally closed and htM = 2, (iii) R = K[X; S] is a monoid domain over a field K, where S = S ∪ {0} is a cancellative torsion-free monoid such that ⋂∞ m=1 mS = ∅, and M is the maximal ideal (Xs : s ∈ S). We extend the above results to ideals I of a reduced ring R such that R/I is Noetherian. We prove that a reduced ring R is Noetherian if each prime ideal of R has a power that is finitely generated. For each d with 3 ≤ d ≤ ∞, we establish existence of a d-dimensional integral domain having a nonfinitely generated maximal ideal M of height d such that M2 is 3-generated.

On linear recursions with nonnegative coefficients

Abstract We characterize here linear recursions which imply a linear recursion with nonnegative coefficients.

On the lifting problem for homogeneous ideals in polynomial rings

We prove here that if k is a field of zero characteristic, then any homogenous ideal in k[X, Y] is liftable to a radical ideal. On the other hand, if k is a finite field, then for any n ≥ 2, there exist zero-dimensional monomial ideals in k[X1,…,Xn] which are not liftable to radical ideals.

When is a power series ring n-root closed?

Abstract Given commutative rings A ⊆ B , we present a necessary and sufficient condition for the power series ring A [[ X ]] to be n -root closed in B [[ X ]]. This result leads to a criterion for the the power series ring A [[ X ]] over an integral domain A to be n -root closed (in its quotient field). For a domain A , we prove: if A is Mori (for example, Noetherian), then A [[ X ]] is n -root closed iff A is n -root closed; if A is Prufer, then A [[ X ]] is root closed iff A is completely integrally closed.