Classical Quotient Ring Research Articles

Extensions of strongly reflexive rings

This is a further study of reflexive rings over polynomial rings and monoid rings. The concepts of strongly reflexive rings and strongly [Formula: see text]-reflexive rings are introduced and investigated. Some characterizations of various extensions of the two classes of rings are obtained. It is proved that a ring [Formula: see text] is strongly reflexive if and only if [Formula: see text] is strongly reflexive if and only if [Formula: see text] is strongly reflexive. For a right Ore ring [Formula: see text] with classical right quotient ring [Formula: see text], we show that [Formula: see text] is strongly reflexive if and only if [Formula: see text] is strongly reflexive. Moreover, we prove that if [Formula: see text] is a unique product monoid (u.p.-monoid) and [Formula: see text] is a reduced ring, then [Formula: see text] is strongly [Formula: see text]-reflexive. It is shown that finite direct sums of strongly [Formula: see text]-reflexive rings are strongly [Formula: see text]-reflexive.

Connectedness of some rings of~quotients of $C(X)$ with the $m$-topology

In this article we define the $m$-topology on some rings of quotients of $C(X)$. Using this, we equip the classical ring of quotients $q(X)$ of $C(X)$ with the $m$-topology and we show that $C(X)$ with the $r$-topology is in fact a subspace of $q(X)$ with the $m$-topology. Characterization of the components of rings of quotients of $C(X)$ is given and using this, it turns out that $q(X)$ with the $m$-topology is connected if and only if $X$ is a pseudocompact almost $P$-space, if and only if $C(X)$ with $r$-topology is connected. We also observe that the maximal ring of quotients $Q(X)$ of $C(X)$ with the $m$-topology is connected if and only if $X$ is finite. Finally for each point $x$, we introduce a natural ring of quotients of $C(X)/O_x$ which is connected with the $m$-topology.

Invariant subrings and Jacobson radicals of Noetherian Hopf module algebras

In this paper several classical facts known for group actions and group gradings on rings are extended to the case of a Noetherian H-module algebra A for a Hopf algebra H. When H is semisimple, a version of the Bergman-Isaacs result is proved, asserting the nilpotency of any onesided ideal of A whose intersection with the subalgebra of H-invariant elements AH is nilpotent. Under the additional assumption that A is H-semiprime, it is established that the classical quotient ring Q(A) of A is the Ore localization of A at the set of H-invariant regular elements. When H is finite-dimensional cosemisimple, the Jacobson radical of A is shown to be stable under the action of H. More generally, these results are obtained for algebras over an arbitrary commutative base ring under suitable restrictions on the Hopf algebra and its action.

The classical ring of quotients of $C_c(X)$

We construct the classical ring of quotients of the algebra of continuous real-valued functions with countable range. Our construction is a slight modification of the construction given in [M. Ghadermazi, O.A.S. Karamzadeh, and M. Namdari, On the functionally countable subalgebra of C(X) , Rend. Sem. Mat. Univ. Padova, to appear]. Dowker's example shows that the two constructions can be different.

On classical rings of quotients of duo rings

In this work we will construct a duo ring R such that the classical right ring of quotients Qclr(R) of R is neither right nor left duo.

Properties which do not pass to classical rings of quotients

We construct examples of Ore rings satisfying some standard ring-theoretic properties for which the classical rings of quotients do not satisfy those properties. Examples of properties which do not pass to rings of quotients include: Abelian, Dedekind-finite, semicommutative, 2-primal, and NI. In the process of constructing these examples we correct the literature. We also introduce two important construction techniques.

Nil 3-Armendariz Rings

We introduce nil 3-Armendariz rings, which are generalization of 3-Armendariz rings and nil Armendaiz rings and investigate their properties. We show that a ring R is nil 3-Armendariz ring if and only if for any , Tn(R) is nil 3-Armendariz ring. Also we prove that a right Ore ring R is nil 3-Armendariz if and only if so is Q, where Q is the classical right quotient ring of R. With the help of this result, we can show that a commutative ring R is nil 3-Armendariz if and only if the total quotient ring of R is nil 3-Armendariz.

Martindale rings and H-module algebras with invariant characteristic polynomials

Under study is the category Open image in new window of the possibly noncommutative H-module algebras that are mapped homomorphically onto commutative algebras. The H-equivariant Martindale ring of quotients QH(A) is shown to be a finite-dimensional Frobenius algebra over the subfield of invariant elements QH(A)H and also the classical ring of quotients for A. We introduce a full subcategory Open image in new window of Open image in new window such that the algebras in Open image in new window are integral over its subalgebras of invariants and construct a functor Open image in new window → Open image in new window, which is left adjoined to the inclusion Open image in new window → Open image in new window.

PROPERTIES OF K-RINGS AND RINGS SATISFYING SIMILAR CONDITIONS

Jacobson introduced the concept of K-rings, continuing the investigation of Kaplansky and Herstein into the commutativity of rings. In this note we focus on the ring-theoretic properties of K-rings. We first construct basic examples of K-rings to be handled easily. It is shown that a semiprime K-ring of bounded index of nilpotency is a commutative domain. It is proved that if R is a prime K-ring then its classical quotient ring is a local ring with a nil Jacobson radical. We also show that if R is a π-regular K-ring then R/P is a field for every strongly prime ideal P of R. The basic structure of a condition, unifying K-rings and reversible rings, is studied with respect to zero-divisors in matrices and polynomials.

A CHARACTERIZATION OF CERTAIN MORPHIC TRIVIAL EXTENSIONS

Given a ring R, we study the bimodules M for which the trivial extension R ∝ M is morphic. We obtain a complete characterization in the case where R is left perfect, and we prove that R ∝ Q/R is morphic when R is a commutative reduced ring with classical ring of quotients Q. We also extend some known results concerning the connection between morphic rings and unit regular rings.

The McCoy Condition on Noncommutative Rings

McCoy proved in 1957 [12] that if a polynomial annihilates an ideal of polynomials over any ring then the ideal has a nonzero annihilator in the base ring. We first elaborate this McCoy's famous theorem further, expanding the inductive construction in the proof given by McCoy. From the proof we can naturally find nonzero c, with f(x)c = 0, in the ideal of R generated by the coefficients of g(x), when f(x), g(x) are nonzero polynomials over a commutative ring R with f(x)g(x) = 0; from which we also obtain a kind of criterion for given a polynomial to be a zero divisor. Based on these results we extend the McCoy's theorem to noncommutative rings, introducing the concept of strong right McCoyness. The strong McCoyness is shown to have a place between the reversibleness (right duoness) and the McCoyness. We introduce a simple way to construct a right McCoy ring but not strongly right McCoy, from given any (strongly) right McCoy ring. If given a ring is reversible or right duo, then the polynomial ring over it is proved to be strongly right McCoy. It is shown that the (strong) right McCoyness can go up to classical right quotient rings.

On a Hereditary Radical Property Relating to the Reducedness

In this note, a hereditary radical property, called homomorphically reduced rings, is introduced, observed, and applied. The dual concept of this property is also studied with the help of Courter, proving that any ring R (possibly without identity) has an ideal S such that S/K is not homomorphically reduced for each proper ideal K of S; and if L is an ideal of R with L ⊊ S, then L/H is homomorphically reduced for some ideal H of R with H ⊊ L. The concept of the homomorphical reducedness is shown to be equivalent to the left (right) weak regularity and the (strong) regularity for one-sided duo rings. It is proved that homomorphically reduced rings have several useful properties similar to those of (weakly) regular rings. It is proved that the homomorphical reducedness can go up to classical quotient rings. It is shown that if R is a reduced right Ore ring with the ascending chain condition (ACC) for annihilator ideals, then the maximal right quotient ring of R is strongly regular (hence homomorphically reduced).

ON A GENERALIZATION OF THE MCCOY CONDITION

We in this note consider a new concept, so called <TEX>$\pi$</TEX>-McCoy, which unifies McCoy rings and IFP rings. The classes of McCoy rings and IFP rings do not contain full matrix rings and upper (lower) triangular matrix rings, but the class of <TEX>$\pi$</TEX>-McCoy rings contain upper (lower) triangular matrix rings and many kinds of full matrix rings. We first study the basic structure of <TEX>$\pi$</TEX>-McCoy rings, observing the relations among <TEX>$\pi$</TEX>-McCoy rings, Abelian rings, 2-primal rings, directly finite rings, and (<TEX>$\pi-$</TEX>)regular rings. It is proved that the n by n full matrix rings (<TEX>$n\geq2$</TEX>) over reduced rings are not <TEX>$\pi$</TEX>-McCoy, finding <TEX>$\pi$</TEX>-McCoy matrix rings over non-reduced rings. It is shown that the <TEX>$\pi$</TEX>-McCoyness is preserved by polynomial rings (when they are of bounded index of nilpotency) and classical quotient rings. Several kinds of extensions of <TEX>$\pi$</TEX>-McCoy rings are also examined.

ON FULLY IDEMPOTENT RINGS

We continue the study of fully idempotent rings initiated by Courter. It is shown that a (semi)prime ring, but not fully idempotent, can be always constructed from any (semi)prime ring. It is shown that the full idempotence is both Morita invariant and a hereditary radical property, obtaining <TEX>$hs(Mat_n(R))\;=\;Mat_n(hs(R))$</TEX> for any ring R where hs(-) means the sum of all fully idempotent ideals. A non-semiprimitive fully idempotent ring with identity is constructed from the Smoktunowicz's simple nil ring. It is proved that the full idempotence is preserved by the classical quotient rings. More properties of fully idempotent rings are examined and necessary examples are found or constructed in the process.

The minimal prime spectrum of rings with annihilator conditions

In this paper, we study rings with the annihilator condition (a.c.) and rings whose space of minimal prime ideals, Min ( R ) , is compact. We begin by extending the definition of (a.c.) to noncommutative rings. We then show that several extensions over semiprime rings have (a.c.). Moreover, we investigate the annihilator condition under the formation of matrix rings and classical quotient rings. Finally, we prove that if R is a reduced ring then: the classical right quotient ring Q ( R ) is strongly regular if and only if R has a Property (A) and Min ( R ) is compact, if and only if R has (a.c.) and Min ( R ) is compact. This extends several results about commutative rings with (a.c.) to the noncommutative setting.

Normal Quadratics in Ore Extensions, Quantum Planes, and Quantized Weyl Algebras

Let R be a Noetherian domain and let (σ,δ) be a quasi-derivation of R such that σ is an automorphism. There is an induced quasi-derivation on the classical quotient ring Q of R. Suppose F=t 2−v is normal in the Ore extension R[t;σ,δ] where v∈R. We show F is prime in R[t;σ,δ] if and only if F is irreducible in Q[t;σ,δ] if and only if there does not exist w∈Q such that v=σ(w)w−δ(w). We apply this result to classify prime quadratic forms in quantum planes and quantized Weyl algebras.

Group actions and rational ideals

We develop the theory of rational ideals for arbitrary associative algebras R without assuming the standard finiteness conditions, noetherianness or the Goldie property. The Amitsur-Martindale ring of quotients replaces the classical ring of quotients which underlies the previous definition of rational ideals but is not available in a general setting. Our main result concerns rational actions of an affine algebraic group G on R. Working over an algebraically closed base field, we prove an existence and uniqueness result for generic rational ideals in the sense of Dixmier: for every G-rational ideal I of R, the closed subset of the rational spectrum RatR that is defined by I is the closure of a unique G-orbit in RatR. Under additional Goldie hypotheses, this was established earlier by Mœglin and Rentschler (in characteristic 0) and by Vonessen (in arbitrary characteristic), answering a question of Dixmier.

Hopf Algebra Orbits on the Prime Spectrum of a Module Algebra

For a Hopf algebra H and an H-module algebra A module-finite over its center it is proved that there is an equivalence relation on a subset SpecfA of the prime spectrum of A which exactly corresponds to the orbit relation in case of group actions. A linearly compact topologically H-simple H-module algebra LP(A) have been associated with each \(P\in\mathop{\rm Spec}_fA\). When A is noetherian and H-semiprime, it is shown that A has a quasi-Frobenius classical quotient ring.

CLEAN CLASSICAL RINGS OF QUOTIENTS OF COMMUTATIVE RINGS, WITH APPLICATIONS TO C(X)

Commutative clean rings and related rings have received much recent attention. A ring R is clean if each r ∈ R can be written r = u + e, where u is a unit and e an idempotent. This article deals mostly with the question: When is the classical ring of quotients of a commutative ring clean? After some general results, the article focuses on C(X) to characterize spaces X when Qcl(X) is clean. Such spaces include cozero complemented, strongly 0-dimensional and more spaces. Along the way, other extensions of rings are studied: directed limits and extensions by idempotents.

Marginal isomorphisms

Let G be a right module over a ring R and let Q G denote the semi-primary classical right ring of quotients of End R ( G ) . Modules G and H are margimorphic if there are maps a : G → H and b : H → G such that ab and ba are regular elements in the respective endomorphism rings. The module H is called a marginal summand of G if G is margimorphic to H ⊕ H ′ for some module H ′ . We study the existence and uniqueness of marginal summands of G n for integers n > 0 in terms of finitely generated projective right Q G -modules. Some of these results extend to direct summands of G n for integers n > 0 .