Edgar A. Rutter

Bio: Edgar A. Rutter is an academic researcher from University of Kansas. The author has contributed to research in topics: Von Neumann regular ring & Ideal (set theory). The author has an hindex of 9, co-authored 15 publications receiving 198 citations.

Rings of quotients of endomorphism rings of projective modules.

Abstract: This paper investigates two related problems. The first is to describe the double centralizer of an arbitrary projective right iϋ-modiile. This proves to be the ring of left quotients of R with respect to a certain canonical hereditary torsion class of left ϋί-modules determined by the projective module. The second is to determine the relationship between rings of left quotients of R and S, where S is the endomorphism ring of a finitely generated projective right ϋί-module PR. It is shown that there exists an inclusion-preserving, one-to-one correspondence between hereditary torsion classes (or localizing subcategories) of left ^-modules and hereditary torsion classes of left E-modules which contain the canonical torsion class determined by PR. If QR and Qs are rings of left quotients with respect to corresponding classes, then P($$RQR is a finitely generated projective right Q^-module with Qs as its Q#-endomorphism ring. Necessary and sufficient conditions are obtained for the maximal rings of left quotients to be related in this manner. In particular, this occurs when PR is a faithful .R-module and R is either a semi-prime ring or a ring with zero left singular ideal. The situation considered includes the case where S is an arbitrary ring, SP is a left ^-generator, and R is the Sendomorphism ring of SP When SP is a projective left Sgenerator, the maximal rings of left quotients of R and S are related in the manner considered above.

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

Abstract: 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

A characterizati on of qf-3 rings

Abstract: Let R be a ring with minimum condition on left or right ideals. It is shown that R is a QF-Z ring if and only if each finitely generated submodule of the injective hull of R, regarded as a left ϋί-module, is torsionless. The same approach yields a simplified proof that R is quasi-Frobenius if and only if every finitely generated left i?-module is torsionless. A ring with identity is called a left QF-Z ring if it has a (unique) minimal faithful left module, and a QF-Z ring means a ring which is both left and right QF-Z. This class of rings originated with Thrall [9] as a generalization of quasi-Frobeni us or QF algebras and has been studied extensively in recent years. Quasi-Frobenius rings have many interesting characterizations and in most instances there exists an analogous characterization of QF-Z rings at least in the case of rings with minimum condition and often for a much larger class of rings. It is well known that a ring with minimum condition on left or right ideals is a left QF-Z ring if and only if the injective hull E{BR) of the ring R regarded as a left iϋ-module is protective. Moreover, in this case R is a QF-Z ring (cf. [6] and [8]). For semi-primary or perfect rings; however, the situation is somewhat different. Namely, a perfect ring is a left QF-Z ring if and only if E(RR) is torsionless. A module is called torsionless if it can be embedded in a direct product of copies of the ring regarded as a module over itself. In this case E(RR) need not be protective and R need not be right QF-Z (cf. [3] and [8]). However, a perfect ring is QF-Z if and only if both E(RR) and E(RR) are protective (see [8]). In this note, it is shown that if R is left perfect ring, E(RR) is protective if and only if each finitely generated submodule of E{RR) can be embedded in a free i?-module. For a ring with minimum condition on left or right ideals this latter condition is equivalent to each finitely generated submodule of E{BR) being torsionless. Thus in that case QF-Z rings may be characterized by this weaker condition. The technique of proof also yields a much simplified proof of a characterization of QF rings given by the present author in [7]. Namely, a ring with minimum condition on left or right ideals is QF if and only if each finitely generated left module is torsionless. Indeed, the characterization of QF-Z rings given here may be regarded as the analog of that result.

Armendariz rings and gaussian rings

Abstract: We prove a number of results concerning Armendariz rings and Gaussian rings. Recall that a (commutative) ring R is (Gaussian) Armendariz if for two polynomials f,g∈R[X] (the ideal of R generated by the coefficients of f g is the product of the ideals generated by the coefficients of f and g) fg = 0 implies a i b j=0 for each coefficient a i of f and b j of g. A number of examples of Armendariz rings are given. We show that R Armendariz implies that R[X] is Armendariz and that for R von Neumann regularR is Armendariz if and only if R is reduced. We show that R is Gaussian if and only if each homomorphic image of R is Armendariz. Characterizations of when R[X] and R[X] are Gaussian are given.

Quasi-Frobenius Rings

Abstract: A ring is called quasi-Frobenius if it is right or left selfinjective, and right or left artinian (all four combinations are equivalent). The study of these rings grew out of the theory of representations of a finite group as a group of matrices over a field, and the subject is intimately related to duality, the duality from right to left modules induced by the hom functor and the duality related to annihilators. The present extent of the theory is vast, and this book makes no attempt to be encyclopedic; instead it provides an elementary, self-contained account of the basic facts about these rings at a level allowing researchers and graduate students to gain entry to the field.

Multiplication Ideals, Multiplication Rings, and the Ring R(X)

Abstract: Let R be a commutative ring with an identity. An ideal A of R is called a multiplication ideal if for every ideal B ⊆ A there exists an ideal C such that B = AC. A ring R is called a multiplication ring if all its ideals are multiplication ideals. A ring R is called an almost multiplication ring if RM is a multiplication ring for every maximal ideal M of R. Multiplication rings and almost multiplication rings have been extensively studied—for example, see [4; 8; 9; 11; 12; 15; and 16].

The quotient category of a Morita context

Abstract: Every Morita context between rings R and S leads to an equivalence between two quotient categories of the module categories mod R and mod S . As consequences, one obtains a generalization of the Morita Theorems, and one constructs induced contexts between quotient rings of R and S . The concept of context-equivalence of rings is introduced and studied. The last part reviews and reorganizes various topics utilizing the new notions and results.