The content of a Gaussian polynomial is invertible

Abstract

Abstract. Let R be an integral domain and let f (X) be a nonzero polynomial in R [X].

The weak dimensions of Gaussian rings

“…Both definitions first appeared in Tsang's thesis [12], where it is also proved that polynomials with invertible, or more generally locally principal, content ideals are Gaussian polynomials. The question whether the converse holds received a great deal of attention in recent years; Glaz & Vasconcelos [6,7], Heinzer & Huneke [8], Loper & Roitman [9], and Lucas [10]. [3] provides a survey of the results obtained until the year 2000, and an extensive reference list.…”

Gaussian polynomials and invertibility

“…The next lemma and its proof are embedded in the proof of [LR,Lemma 2]. Note that Lemma 2 of [LR] is stated in terms of quasilocal integral domains, while our Lemma 3 and Lemma 4 deal with arbitrary commutative rings.…”

“…As with Lemma 3, the statement and the essence of the proof are from the proof given for [LR,Lemma 2].…”

“…By way of the following example, we show that the converse does not hold. [Note: The converse does hold for integral domains [G,Theorem 28.6] (and now also by [LR,Theorem 4]). ] In some sense this example generalizes the statement following Theorem 1.1 in [AK] -if R is a quasilocal ring with a nonzero maximal ideal whose square is zero, then each polynomial over R is Gaussian.…”

Commutative ideal theory without finiteness conditions: Completely irreducible ideals

“…Our goal in this paper is to examine the structure of completely irreducible ideals of a commutative ring on which there are imposed no finiteness conditions. Other recent papers that address the structure and ideal theory of rings without finiteness conditions include [3], [4], [8], [10], [14], [15], [16], [19], [25], [26].…”