Class Of Commutative Rings Research Articles

$HSP\not = SHPS$ for commutative rings with identity

Let $I$, $H$, $S$, $P$, $P_s$ be the usual operators on classes of rings: $I$ and $H$ for isomorphic and homomorphic images of rings and $S$, $P$, $P_s$ respectively for subrings, direct, and subdirect products of rings. If $\mathcal K$ is a class of commutative rings with identity (and in general of any kind of algebraic structures), then the class $HSP({\mathcal K})$ is known to be the variety generated by the class $\mathcal K$. Although the class $SHPS({\mathcal K})$ is in general a proper subclass of the class $HSP({\mathcal K})$ for many familiar varieties $HSP({\mathcal K})= SHPS({\mathcal K})$. Our goal is to give an example of a class $\mathcal K$ of commutative rings with identity such that $HSP({\mathcal K})\not = SHPS({\mathcal K})$. As a consequence we will describe the structure of two partially ordered monoids of operators.

Primeness Conditions on the Lattice of Hereditary Pretorsion Classes#

A lattice ordered monoid is a structure ⟨L;⊕, 0 L ; ≤⟩ where ⟨L;⊕, 0 L ⟩ is a monoid, ⟨L; ≤ ⟩ is a lattice and the binary operation ⊕ distributes over finite meets. If R is an arbitrary ring with identity, then the set ptors R of all hereditary pretorsion classes in the category of right R-modules is an example of a lattice ordered monoid. It is known that right strongly prime and right strongly semiprime rings are characterizable by means of first order sentences in the language of ptors R. A notion of weak primeness and weak semiprimeness is introduced. These are defined by means of first order sentences in ptors R. It is shown that in the presence of a weak commutativity condition, which is also defined in the language of ptors R, the notions of weak primeness [resp. weak semiprimeness] and strong primeness [resp. strong semiprimeness] coincide. It is also proved that within the class of commutative rings, Quasi–Frobenius rings are characterizable by means of a simple sentence in ptors R.

Modules whose Injective Endomorphisms Are Essential

An R-module M is called weakly co-Hopfian if any injective endomorphism of M is essential. The class of weakly co-Hopfian modules lies properly between the class of co-Hopfian and the class of Dedekind finite modules. Several equivalent conditions are given for a module to be weakly co-Hopfian. Being co-Hopfian, weakly co-Hopfian, or Dedekind finite are all equivalent conditions on quasi-injective modules. Some other properties of weakly co-Hopfian modules are also obtained. The ring R is said to be right strong stably finite if all the finitely generated free right R-modules are weakly co-Hopfian. We shall characterize such rings and show that they are stably finite and satisfy the right strong rank condition. Examples show that stably finite rings and rings with the right strong rank conditions need not be strong stably finite. Both weakly co-Hopfian and right strong stably finite are Morita invariants, although the right and left strong stably finite are different properties. The class of commutative rings and the class of rings with finite right uniform dimension are proper subclasses of the class of right strong stably finite rings. We shall also investigate conditions that are relevant to weakly co-Hopfian modules. Equivalent statements are found on a ring to have all its finitely generated right modules weakly co-Hopfian.

On a problem of Wenzel on equational compactness of modules

In this note, we introduce a weak form of equational compactness of modules, which we called x0-compactness, together with some of its basic properties. We then use it to settle in the negative a problem of G. H. Wenzel. We also prove the adjacent result that, over a wide class of commutative rings, if every module is x0-compact then the ring has finite representation type.

Commutative rings with comparable regular elements

AbstractLet ℜbe the class of commutative ringsRwith comparable regular elements, that is, given two non zero-divisors inR, one divides the other. Applying the notion ofV-valuation due to Harrison and Vitulli, we define the classV-val ofV-valuated rings, which is contained in ℜand contains the class of Manis valuation rings. We prove that these inclusions of classes are both proper. We investigate Prüfer rings inside ℜ, showing that there exist Prüfer rings which lie in ℜbut not inV-val; we prove that a ringRis a Prüfer valuation ring if and only if it is Prüfer andV-valuated, if and only if its lattice of regular ideals is a chain. Finally, we introduce and investigate the idealI∞of a ringR∈ ℜ, which corresponds to the counterimage of ∞, wheneverRisV-valuated.

A class of commutative rings

A class of commutative rings

Radical classes of commutative rings defined in terms of the adjoint operation

Radical classes of commutative rings defined in terms of the adjoint operation

Totally Integrally Closed Azumaya Algebras

AbstractEnochs introduced and studied totally integrally closed rings in the class of commutative rings. This article studies the same question for Azumaya algebras, a study made possible by Atterton's notion of integral extensions for non-commutative rings.The main results are that Azumaya algebras are totally integrally closed precisely when their centres are, and that an Azumaya algebra over a commutative semiprime ring has a tight integral extension that is totally integrally closed. Atterton's integrality differs from that often studied but is very natural in the context of Azumaya algebras. Examples show that the results do not carry over to free normalizing or excellent extensions.

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

An Answer to a Question of M. Newman on Matrix Completion

Let $R$ be a principal ideal ring, $A$ a symmetric $t$-by-$t$ matrix over $R$, $B$ a $t$-by-$(n - t)$ matrix over $R$ such that the $t$-by-$n$ matrix $(A,B)$ is primitive. Newman [2] proved that $(A,B)$ may be completed (as the first $t$ rows) to a symmetric $n$-by-$n$ matrix of determinant 1, provided that $1 \leq t \leq n/3$. He showed that the result is false, in general, if $t = n/2$, and he asked to determine all values of $t$ such that $1 \leq t \leq n$ and the result holds. It is shown here that these values are exactly $t$ satisfying $1 \leq t \leq n/2$. Moreover, the result is proved for a larger (than the principal ideal rings) class of commutative rings, namely, for the rings satisfying the second stable range condition of Bass [1]. Also, it is observed that Theorems 2 and 3 of [2, p. 40] proved there for principal ideal rings are true for this larger class of rings, as well as the basic result of [2, p. 39].

An answer to a question of M. Newman on matrix completion

Let R R be a principal ideal ring, A A a symmetric t t -by- t t matrix over R R , B B a t t -by- ( n − t ) (n - t) matrix over R R such that the t t -by- n n matrix ( A , B ) (A,B) is primitive. Newman [2] proved that ( A , B ) (A,B) may be completed (as the first t t rows) to a symmetric n n -by- n n matrix of determinant 1, provided that 1 ≤ t ≤ n / 3 1 \leq t \leq n/3 . He showed that the result is false, in general, if t = n / 2 t = n/2 , and he asked to determine all values of t t such that 1 ≤ t ≤ n 1 \leq t \leq n and the result holds. It is shown here that these values are exactly t t satisfying 1 ≤ t ≤ n / 2 1 \leq t \leq n/2 . Moreover, the result is proved for a larger (than the principal ideal rings) class of commutative rings, namely, for the rings satisfying the second stable range condition of Bass [1]. Also, it is observed that Theorems 2 and 3 of [2, p. 40] proved there for principal ideal rings are true for this larger class of rings, as well as the basic result of [2, p. 39].

Uniqueness of barbilian domains

W. Leissner has developed a plane geometry over any Z-ring R, in which a point is an element of R×R and a line is a set of the form {(x+ ra, y + rb):r ∈ R} where (x,y) ∈ R×R and (a,b) is from a “Barbilian domain”, i.e., a set of unimodular pairs from R×R satisfying certain axioms. In this note we generalize results of W. Benz guaranteeing the uniqueness of Barbilian domains over several classes of commutative rings. The author wishes to thank Gordon Keller and Douglas Costa for fruitful discussions, the referee for his improvements, and the University of Virginia for its hospitality while this work was done.

Amalgamation bases for commutative rings without nilpotent elements

SCR=the class of commutative rings without nilpotent elements.Theorem,R is an amalgamation base for SCR iff rad (I=Ann2(I) forI ⊆R finitely generated.

A note on commutative Baer rings

The class of commutative rings known as Baer rings was first discussed by J. Kist [4], where many interesting properties of these rings were established. Not necessarily commutative Baer rings had previously been studied by I. Kaplansky [3], and by R. Baer himself [1]. In this note we show that commutative Baer rings, which generalize Boolean rings and p-rings, satisfy the Birkhoff conditions for a variety. Next we give a set of equations characterising this variety involving + and * as binary operations, – and as unary operations, and 0 as nullary operation. Finally we describe Baer-subdirectly irreducible commutative Baer rings and state the appropriate representation theorem.

Inductive ring topologies

Since topological algebra is the study of algebraic structures with topologies for which the operations are continuous, a natural question for the topological algebraist to ask is whether a given structure admits any such topologies whatever, other than the discrete and indiscrete ones. The question has been answered for some classes of structures. For example, Kertesz and Szele [7] prove that every infinite abelian group admits a nondiscrete, Hausdorff group On the other hand, Hanson [5] gives an example of an infinite groupoid which admits only the two trivial topologies mentioned above. Our purpose here is to answer this question for infinite fields, proving that every infinite field admits a nondiscrete, Hausdorff field This will be done by affirmatively answering the question for two classes of commutative rings: the first being all integral domains with a certain cardinality condition (§3), and the second, all rings which are the union of a chain of subrings with certain properties (§4). These two classes will be shown to include all infinite fields (§5). Our method of proof will make use of an procedure first used by Hinrichs [6] to prove the existence of certain unusual topologies on the integers. The procedure is described in §1, where we define what we mean by an inductive ring topology. In §§7 and 8, we turn our attention to some further applications of topologies, showing first how they can be used to construct interesting examples of topologies on the integers and rational numbers. We use them to get proofs that there are uncountably many, and non-first countable ring topologies on all the rings considered in §3 and §4. We also show how characterizations can be obtained for several classes of topologies on fields using modifications of the method. A supplement to our discussion of field topologies comes in §6, where we characterize those fields which admit nondiscrete, Hausdorff, locally bounded topologies. The methods used here, however, are those of valuation theory. When we say that a topology S~ is a ring topology on a ring A, we mean that the mappings (a, b)^-a—b and (a, b)-*ab from Ax A into A are continuous. &~ is a field topology on a field K if it is a ring topology, and in addition, the mapping a -*■ a'1 is continuous on K~{0}.