Ring-theoretic Properties Research Articles

The length function and matrix algebras

By the length of a finite system of generators for a finite-dimensional associative algebra over an arbitrary field we mean the least positive integer k such that words of length not exceeding k span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. In the present paper, we study the main ring-theoretical properties of the length function: the behavior of the length under unity adjunction, direct sum of algebras, passage to subalgebras and homomorphic images. We give an upper bound for the length of the algebra as a function of the nilpotency index of its Jacobson radical and the length of the quotient algebra. We also provide examples of length computation for certain algebras, in particular, for the following classical matrix subalgebras: the algebra of upper triangular matrices, the algebra of diagonal matrices, the Schur algebra, Courter’s algebra, and for the classes of local and commutative algebras.

Radicals of Skew Inverse Laurent Series Rings

In this note, we continue to study zero-divisor properties of skew inverse Laurent series rings R((x −1; σ, δ)), where R is an associative ring equipped with an automorphism σ and a σ-derivation δ. We first introduce (σ, δ)-SILS Armendariz rings, a generalization of the standard Armendariz condition from ordinary polynomial ring R[x] to skew inverse Laurent series ring R((x −1; σ, δ)). We study the ring-theoretical properties of (σ, δ)-SILS Armendariz rings and using the properties of these rings, we characterize radicals of the skew inverse Laurent series ring R((x −1; σ, δ)), in terms of a (σ, δ)-SILS Armendariz ring R. We also prove that several properties transfer between R and R((x −1; σ, δ)), in case R is an σ-compatible (σ, δ)-SILS Armendariz ring.

THE INTERSECTION GRAPH OF GAMMA SETS IN THE TOTAL GRAPH OF A COMMUTATIVE RING-I

Let R be a commutative ring and Z(R) be its set of all zero-divisors. Anderson and Badawi [The total graph of a commutative ring, J. Algebra320 (2008) 2706–2719] introduced the total graph of R, denoted by TΓ(R), as the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Tamizh Chelvam and Asir [Domination in the total graph of a commutative ring, to appear in J. Combin. Math. Combin. Comput.] obtained the domination number of the total graph and studied certain other domination parameters of TΓ(R) where R is a commutative Artin ring. The intersection graph of gamma sets in TΓ(R) is denoted by ITΓ(R). Tamizh Chelvam and Asir [Intersection graph of gamma sets in the total graph, Discuss. Math. Graph Theory32 (2012) 339–354, doi:10.7151/dmgt.1611] initiated a study about the intersection graph ITΓ (ℤn) of gamma sets in TΓ(ℤn). In this paper, we study about ITΓ(R), where R is a commutative Artin ring. Actually we investigate the interplay between graph-theoretic properties of ITΓ(R) and ring-theoretic properties of R. At the first instance, we prove that diam (ITΓ(R)) ≤ 2 and gr (ITΓ(R)) ≤ 4. Also some characterization results regarding completeness, bipartite, cycle and chordal nature of ITΓ(R) are given. Further, we discuss about the vertex-transitive property of ITΓ(R). At last, we obtain all commutative Artin rings R for which ITΓ(R) is either planar or toroidal or genus two.

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.

Decomposition of monomial algebras: Applications and algorithms

Considering finite extensions K(A) K(B) of positive affine semigroup rings over a field K we have developed in (BEN) an algorithm to decompose K(B) as a direct sum of monomial ideals in K(A). By computing the regularity of homogeneous semigroup rings from the decomposition, we have confirmed the Eisenbud-Goto conjecture in a range of new cases not tractable by standard methods. Here we first illustrate this technique and its implementation in our Macaulay2 package MonomialAlgebras by computing the decomposition and the regularity step by step for an explicit example. We then focus on ring-theoretic properties of simplicial semigroup rings. From the characterizations given in (BEN), we develop and prove explicit algorithms testing various properties, including being Buchsbaum, Cohen- Macaulay, Gorenstein, normal, and seminormal. All algorithms are implemented in our Macaulay2 package.

A correspondence between the isobaric ring and multiplicative arithmetic functions

We give a representation of the classical theory of multiplicative arithmetic functions (MF) in the ring of symmetric polynomials written on the isobaric basis. The representing elements are recursive sequences of Schur-hook polynomials evaluated on subrings of the complex numbers. The multiplicative arithmetic functions are units in the Dirichlet ring of arithmetic functions, and their properties can be described locally, that is, at each prime number p. Our representation is, hence, a local representation. This representation enables us to clarify and generalize classical results, e.g., the Busche-Ramanujan identity, as well as to give a richer structural description of the convolution group of multiplicative functions. It is a consequence of the representation that the MFs can be defined in a natural way on the negative powers of the prime p which, in turn, leads to a natural extension of Schur-hook polynomials to negatively indexed Schur-hook polynomials. CONTENTS 0. Introduction 1. Ring of Isobaric Polynomials 2. The Companion Matrix and Core Polynomials 3 The Ring of Arithmetic Functions 4. Relation Between Arithmetic Functions and the WIP-Module 5. Examples 6. Specially Multiplicative Arithmetic Functions 7. Structure of the Convolution Group of Multiplicative Arithmetic Functions Reconsidered 8. Norms 0. INTRODUCTION In this paper we give a representation of the classical theory of multiplicative arithmetic functions (MF) in the ring of symmetric polynomials. The Dirichlet ring of arithmetic functions A∗ is well known to be a unique factorization domain.(see Cashwell and Everett [?]). Its ring theoretic properties have been investigated in, e.g., Rearick [?] and [?], Shapiro [?], Carroll and Gioia [?], MacHenry [?], MacHenry and Tudose [?]. The multiplicative arithmetic functions are units in this ring and their properties can be described locally, that is, at each prime number p, (see, e.g., McCarthy [?], Sivaramakishnan[?] and Vaidyanathswamy, [?]). It is this local behaviour which we take advantage of to construct a representation in terms of a certain class of symmetric polynomials

On Rings Having McCoy-Like Conditions

In [41], Nielsen proves that all reversible rings are McCoy and gives an example of a semicommutative ring that is not right McCoy. At the same time, he also shows that semicommutative rings do have a property close to the McCoy condition. In this article we study weak McCoy rings as a common generalization of McCoy rings and weak Armendariz rings. Relations between the weak McCoy property and other standard ring theoretic properties is considered. We also study the weak skew McCoy condition, a generalization of the standard weak McCoy condition from polynomials to skew polynomial rings. We resolve the structure of weak skew McCoy rings and obtain various necessary or sufficient conditions for a ring to be weak skew McCoy, unifying and generalizing a number of known McCoy-like conditions in the special cases. Constructing various examples, we classify how the weak McCoy property behaves under various ring extensions. As a consequence we extend and unify several known results related to McCoy rings and Armendariz rings [6, 35, 38, 43, 49].

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.

Decomposition of Semigroup Algebras

Let A⊆B be cancellative abelian semigroups, and let R be an integral domain. We show that the semigroup ring R[B] can be decomposed, as an R[A]-module, into a direct sum of R[A]-submodules of the quotient ring of R[A]. In the case of a finite extension of positive affine semigroup rings, we obtain an algorithm computing the decomposition. When R[A] is a polynomial ring over a field, we explain how to compute many ring-theoretic properties of R[B] in terms of this decomposition. In particular, we obtain a fast algorithm to compute the Castelnuovo–Mumford regularity of homogeneous semigroup rings. As an application we confirm the Eisenbud–Goto conjecture in a range of new cases. Our algorithms are implemented in the Macaulay2 package MonomialAlgebras.

RINGS OVER WHICH COEFFICIENTS OF NILPOTENT POLYNOMIALS ARE NILPOTENT

Antoine studied conditions which are connected to the question of Amitsur of whether or not a polynomial ring over a nil ring is nil, observing the structure of nilpotent elements in Armendariz rings and introducing the notion of nil-Armendariz rings. The class of nil-Armendariz rings contains Armendariz rings and NI rings. We continue the study of nil-Armendariz rings, concentrating on the structure of rings over which coefficients of nilpotent polynomials are nilpotent. In the procedure we introduce the notion of CN-rings that is a generalization of nil-Armendariz rings. We first construct a CN-ring but not nil-Armendariz. This may be a base on which Antoine's theory can be applied and elaborated. We investigate basic ring theoretic properties of CN-rings, and observe various kinds of CN-rings including ordinary ring extensions. It is shown that a ring R is CN if and only if R is nil-Armendariz if and only if R is Armendariz if and only if R is reduced when R is a von Neumann regular ring.

Normal Pairs of Going-down Rings

Let (R, T) be a normal pair of commutative rings (i.e., R T is a unita extension of commutative rings, not necessarily integral domains, such that S is integrally closed in T for each ring S such that R S T) such that the total quotient ring of R is a von Neumann regular ring. Let P be one of the following ring-theoretic properties: going-down ring, extensionally going-down (EGD) ring, locally ring. Then R has P and only T has P. An example shows that the if part of the assertion fails P is taken to be the divided domain property.

A Kind of Graph Structure on Non-reduced Rings

In this paper, a kind of graph structure ΓN(R) of a ring R is introduced, and the interplay between the ring-theoretic properties of R and the graph-theoretic properties of ΓN(R) is investigated. It is shown that if R is Artinian or commutative, then ΓN(R) is connected, the diameter of ΓN(R) is at most 3; and if ΓN(R) contains a cycle, then the girth of ΓN(R) is not more than 4; moreover, if R is non-reduced, then the girth of ΓN(R) is 3. For a finite commutative ring R, it is proved that the edge chromatic number of ΓN(R) is equal to the maximum degree of ΓN(R) unless R is a nilpotent ring with even order. It is also shown that, with two exceptions, if R is a finite reduced commutative ring and S is a commutative ring which is not an integral domain and ΓN(R) ≃ ΓN(S), then R ≃ S. If R and S are finite non-reduced commutative rings and ΓN(R) ≃ ΓN(S), then |R|=|S| and |N(R)|=|N(S)|.

Concerning P-frames, essential P-frames, and strongly zero-dimensional frames

We give characterizations of P-frames, essential P-frames and strongly zero-dimensional frames in terms of ring-theoretic properties of the ring of continuous real-valued functions on a frame. We define the m-topology on the ring \({\mathcal{R}L}\) and show that if L belongs to a certain class of frames properly containing the spatial ones, then L is a P-frame iff every idealof \({\mathcal{R}L}\) is m-closed. We define essential P-frames (analogously to their spatial antecedents) and show that L is a proper essential P frame iff all the nonmaximal prime ideals of \({\mathcal{R}L}\) are contained in one maximal ideal. Further, we show that L is strongly zero-dimensional iff \({\mathcal{R}L}\) is a clean ring, iff certain types of ideals of \({\mathcal{R}L}\) are generated by idempotents.

ZERO-DIVISOR GRAPHS WITH RESPECT TO PRIMAL AND WEAKLY PRIMAL IDEALS

Abstract. We consider zero-divisor graphs with respect to primal, non-primal, weakly prime and weakly primal ideals of a commutative ring Rwith non-zero identity. We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Γ I (R) forsome ideal I of R. Also we show that the zero-divisor graph with respectto primal ideals commutes by localization. 1. IntroductionThe idea of associating a graph with the zero-divisors of a commutative ringwas introduced by Beck in 1988, where the author talked about the colorings ofsuch graphs. By the definition he gave, every element of the ring R was a vertexin the graph, and two vertices x,y were adjacent if and only if xy = 0 ([4]).We adopt the approach used by D. F. Anderson and P. S. Livingston ([2]) andconsider only non-zero zero-divisors as vertices of the graph. The zero-divisorgraph of a commutative ring has been studied extensively by several authors(see, for example, [2, 4, 5, 10, 11, 12]).Redmond[13]introduced the definition ofthe zero-divisorgraphwith respectto an ideal. Let I be an ideal of a ring R. The zero-divisor graph of Rwith respect to I is an undirected graph, denoted by Γ

Some ring-theoretic properties of almost P-frames

Almost P-frames generalize almost P-spaces, and indeed transcend them. In this article we give several characterizations of these frames, mostly in terms of certain ring-theoretic properties of \({\mathcal{R}}L\), the ring of real-valued functions on L.

Notes on Pointfree Disconnectivity with a Ring-theoretic Slant

We give characterizations of extremally disconnected frames, basically disconnected frames and F-frames L in terms of ring-theoretic properties of the ring \(\mathcal{R}L\) of continuous real-valued functions on L. Emanating from these are new (and purely ring-theoretic) proofs that a frame is extremally disconnected, basically disconnected or an F-frame iff the same holds for its Cech-Stone compactification.

Malcev rings

In the paper, Malcev rings are defined. The class of Malcev rings strictly contains rings of Malcev–Neumann series, formal skew Laurent series rings, and formal pseudo-differential operator rings. In the paper, ring-theoretic properties of Malcev rings are studied. It turns out that rings of Malcev–Neumann series, formal skew Laurent series rings, and formal pseudo-differential operator rings have similar ring-theoretic properties related to the existence of the filtration with respect to the lowest degree of the series.

On the modularity of supersingular elliptic curves over certain totally real number fields

We study generalisations to totally real fields of the methods originating with Wiles and Taylor and Wiles [A. Wiles, Modular elliptic curves and Fermat's Last Theorem, Ann. of Math. 141 (1995) 443–551; R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. 141 (1995) 553–572]. In view of the results of Skinner and Wiles [C. Skinner, A. Wiles, Nearly ordinary deformations of irreducible residual representations, Ann. Fac. Sci. Toulouse Math. (6) 10 (2001) 185–215] on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction. Combining these, we then obtain some partial results on the modularity problem for semistable elliptic curves, and end by giving some applications of our results, for example proving the modularity of all semistable elliptic curves over Q ( 2 ) .

McCoy rings and zero-divisors

We investigate relations between the McCoy property and other standard ring theoretic properties. For example, we prove that the McCoy property does not pass to power series rings. We also classify how the McCoy property behaves under direct products and direct sums. We prove that McCoy rings with 1 are Dedekind finite, but not necessarily Abelian. In the other direction, we prove that duo rings, and many semi-commutative rings, are McCoy. Degree variations are defined, studied, and classified. The McCoy property is shown to behave poorly with respect to Morita equivalence and (infinite) matrix constructions.

Gelfand Factor Rings and Weak Zariski Topologies

Let R be any ring with identity. Let N(R) (resp. J(R)) denote the prime radical (resp. Jacobson radical) of R, and let Spec r (R) (resp. Spec l (R), Max r (R), Prim r (R)) denote the set of all right prime ideals (resp. all left prime ideals, all maximal right ideals, all right primitive ideals) of R. In this article, we study the relationships among various ring-theoretic properties and topological conditions on Spec r (R) (with weak Zariski topology). The following results are obtained: (1) R/N(R) is a Gelfand ring if and only if Spec r (R) is a normal space if and only if Spec l (R) is a normal space; (2) R/J(R) is a Gelfand ring if and only if every right prime ideal containing J(R) is contained in a unique maximal right ideal.