J.W Brewer

Pole assignability in polynomial rings, power series rings, and Prüfer domains

The topics treated in this paper have their origins in the area of algebraic systems theory. However, the paper itself should be classified as pure commutative algebra and we shall present it as such in the body of the text. Still, it is appropriate to give a brief paragraph of motivation. If a physical system is governed by a pair (F, G) of matrices, then the stability of the system can be determined by examining the eigenvalues of the matrix F. If the system is unstable, a “feedback” matrix K can sometimes be employed in such a way that the eigenvalues of the matrix F+ GK measure the stability of the (modified) system (F+ GK, G). In this manner, an unstable system can be rendered stable. The pole assignability problem over commutative rings is one method of attacking the problem of finding such matrices K. The paper is divided into four sections. Section 1 is given over almost entirely to defining the properties in which we shall be interested. It concludes with a theorem about residuating and lifting the properties. Section 2 is concerned with the preservation of the properties under polynomial ring and power series ring formation. Section 3 is concerned with “feedback cyclization,” a strong form of pole assignability. Section 4 is concerned with pole assignability over Priifer domains. A complete elaboration of our results must await the introduction of the necessary terminology. For now, we mention the following results in somewhat vague language. If R is a zero-dimensional ring, then the pole assignability problem is solvable in R[X] and in R[ [Xl]. From a ringtheoretic standpoint, almost any class of commutative rings contains members for which the feedback cyclization problem is solvable. On the other hand, if R is a ring with 1 in its stable range, then the feedback cyclization problem is solvable in R if and only if a certain nice matricial property

Multiplicative Ideal Theory in Commutative Algebra

For over forty years, Robert Gilmer’s numerous articles and books have had a tremendous impact on research in commutative algebra. It is not an exaggeration to say that most articles published today in non-Noetherian ring theory, and some in Noetherian ring theory as well, originated in a topic that Gilmer either initiated or enriched by his work. This volume, a tribute to his work, consists of twenty-four articles authored by Robert Gilmer’s most prominent students and followers. These articles combine surveys of past work by Gilmer and others, recent results which have never before seen print, open problems, and extensive bibliographies. In a concluding article, Robert Gilmer points out directions for future research, highlighting the open problems in the areas he considers of importance. Robert Gilmer’s article is followed by the complete list of his published works, his mathematical genealogical tree, information on the writing of his four books, and reminiscences about Robert Gilmer’s contributions to the stimulating research environment in commutative algebra at Florida State in the middle 1960s. The entire collection provides an in-depth overview of the topics of research in a significant and large area of commutative algebra.

Coherence and weak global dimension of R[[X]] when R is von Neumann regular

dimension of R. Very little is known about what can occur except that the weak global dimension must increase by at least one and does increase by exactly one when R[[X]] is coherent. This paper attempts to cast some light on this problem and the closely related question of the stability of coherence under the formation of the power series ring. It is devoted to determining necessary and sufficient conditions on a commutative (von Neumann) regular ring R for the power series ring R[[X]] to b e coherent (equivalently, semihereditary) and also conditions for R[[X]] t o h ave weak global dimension one. Surprisingly, it turns out that R[[X]] h as weak global dimension one precisely when R[[X’j] is a B&out ring so that property is characterized as well. Each of these properties is characterized in several ways both in terms of internal conditions on R and conditions that involve the category of R-modules. Perhaps the conditions which can be most readily verified for a specific ring are expressed in terms of a natural partial order z< which is defined on R by a < b if and only if ab = a2 for a, b in R. For example, it is shown that R[[X]] is coherent if and only if every countable subset of R that forms a chain in the partial order < on R has a least upper bound in R. As further illustrations of our results we mention that R[[Xj] h as weak global dimension one precisely in case R satisfies either of two limited forms of self-injectivity and also in case R satisfies a restricted type of algebraic compactness. These results permit us to give an example that shows R[[X]] can have weak global dimension one without being coherent even when R is a Boolean ring. This settles a question raised by Jensen [7]. Another example is included to illustrate the added complexity that occurs in arbitrary regular rings as compared with Boolean rings. This example also provides a pair of regular rings R contained in S, and hence a faithfully flat extension, such that R[[X]] is not pure in S[[X]]. Thus it gives a negative

On the pole assignability property over commutative rings

This paper is concerned with the pole assignability property in commutative rings. Specifically, a commutative ring R has the pole assignability property iff given an n-dimensional reachable system (F, G) over R and ring elements r1,…,rn ϵ R, there exists a matrix K such that the characteristic polynomial of the matrix F+GK is (X−r1) ∣ (X−rn). The principal theorem of this paper is Theorem 3: Let R be a commutative ring with the property that all rank one projective R-modules are free. Then R has the pole assignability property iff given a reachable system (F, G) there is a unimodular vector in the image of G.

Seminormality in power series rings

Let R be a commutative unitary ring. Recently, several authors have been interested in the preservation of seminormality in passing from R to the polynomial ring R [X, ,..., X,,J. In this paper we give a proof in the case of power series rings and the proof is both short and applicable in the polynomial setting. The chief difficulty lies in proving that “relative” stability is preserved in passage to the power series ring. We briefly recall the pertinent terminology. If R is a ring, then following Swan [3] we say that R is seminormal if whenever b, c E R satisfy b3 = c2 there is an element a E R such that a2 = b and a3 = c. If R c T are rings, then R is said to be seminormal in T if and only if whenever u E T with u*, a3 E R, it follows that a E R. It can be seen that this is equivalent to saying that whenever (x E T with an, cP+ l,..., E R for some positive integer n, it follows that (r E R. Indeed, if R is seminormal in T and if n is taken to be minimal so that on, on+‘,..., E R, then (u”~‘)“, (c?‘)~ E R unless n = 1, in which case u E R, anyway. We refer the reader to [2, 31 for nice discussions of seminormality. We can now state the theorem.

The Pole Assignability Property in Polynomial Rings over GCD-Domains

Several recent papers, among them [BSSV, Tl, T2], have considered the problem of identifying rings with the property of “pole assignability.” (The definition appears below.) In particular, Bumby, Sontag, Sussmann, and Vasconcelos [BSSV] showed that, while a polynomial ring in one indeterminate over a field has this property, the polynomial ring in two indeterminates over the reals, R[x, y], and the polynomial ring in one indeterminate over the integers, Z[x], do not have this property. Then Tannenbaum [Tl, T23 showed that the polynomial ring in two indeterminates over any field does not have this property. The purpose of this note is to unify the proofs of these two facts in results that we hope will be helpful in identifying the pole assignability property (or its absence) in other rings. Let