Divisibility theory in commutative rings: Bezout monoids

A ubiquitous class of lattice ordered semigroups introduced by Bosbach, which we call Bezout monoids, seems to be the appropriate structure for the study of divisibility in various classical rings like GCD domains (including UFD??s), rings of low dimension (including semi-hereditary rings), as well as certain subdirect products of such rings and certain factors of such subdirect products. A Bezout monoid is a commutative monoid S with 0 such that under the natural partial order (for a, b ?? S, a ?? b ?? S ? bS ? aS), S is a distributive lattice, multiplication is distributive over both meets and joins, and for any x, y ?? S, if d = x ?? y and dx ₁ = x then there is a y ₁ ?? S with dy ₁ = y and x ₁ ?? y ₁ = 1. We investigate Bezout monoids by using filters and m-prime filters, and describe all homorphisms between them. We also prove analogues of the Pierce and the Grothendieck sheaf representations of rings for Bezout monoids. The question whether Bezout monoids describe divisibility in Bezout rings (rings whose finitely generated ideals are principal) is still open.     

Central semicommutative rings

A ring R is central semicommutative if ab = 0 implies that aRb ? Z(R) for any a, b ∈ R. Since every semicommutative ring is central semicommutative, we study sufficient condition for central semicommutative rings to be semicommutative. We prove that some results of semicommutative rings can be extended to central semicommutative rings for this general settings, in particular, it is shown that every central semicommutative ring is nil-semicommutative. We show that the class of central semicommutative rings lies strictly between classes of semicommutative rings and abelian rings. For an Armendariz ring R, we prove that R is central semicommutative if and only if the polynomial ring R[x] is central semicommutative. Moreover, for a central semicommutative ring R, it is proven that (1) R is strongly regular if and only if R is a left GP-V′-ring whose maximal essential left ideals are GW-ideals if and only if R is a left GP-V′-ring whose maximal essential right ideals are GW-ideals. (2) If R is a left SF and central semicommutative ring, then R is a strongly regular ring.     

Pseudosimilarity and cancellation of modules

Two square matrices A and B over a ring are pseudosimilar if there exist X, Y, and Z satisfying XAY = B, ZBX = A, and XYX = XZX = X. Hartwig and Hall showed this is equivalent to similarity over a field. This result is extended to rings where free modules satisfy a cancellation property. These include rings R with R/rad R artinian (or more generally rings with one in the stable range) and polynomial rings over Dedekind domains. Furthermore, it is shown for commutative rings that if A and B are pseudosimilar, then diag(A, O_m) and diag(B, O_m) are similar for some m.     

Quasi-duo rings and stable range descent

In a recent paper, the first author introduced a general theory of corner rings in noncommutative rings that generalized the classical theory of Peirce decompositions. This theory is applied here to the study of the stable range of rings upon descent to corner rings. A ring is called quasi-duo if every maximal 1-sided ideal is 2-sided. Various new characterizations are obtained for such rings. Using some of these characterizations, we prove that, if a quasi-duo ring R has stable range ?n, the same is true for any semisplit corner ring of R. This contrasts with earlier results of Vaserstein and Warfield, which showed that the stable range can increase unboundedly upon descent to (even) Peirce corner rings.     

Quasi-Armendariz rings relative to a monoid

For a monoid M, we introduce M-quasi-Armendariz rings which are a generalization of quasi-Armendariz rings, and investigate their properties. The M-quasi-Armendariz condition is a Morita invariant property. The class of M-quasi-Armendariz rings is closed under some kinds of upper triangular matrix rings. Every semiprime ring is M-quasi-Armendariz for any unique product monoid and any strictly totally ordered monoid M. Moreover, we study the relationship between the quasi-Baer property of a ring R and those of the monoid ring R[M]. Every quasi-Baer ring is M-quasi-Armendariz for any unique product monoid and any strictly totally ordered monoid M.     

System-theoretic and algebraic aspects of the rings of stable and proper stable rational functions

A new approach in transfer-function methods for solving a variety of control-theoretic problems is to work with fractional representations over the ring of stable or proper stable rational functions of various transfer matrices. The rings of stable and proper stable rational functions are well known to be Euclidean domains. These rings differ critically from the polynomial ring by the nonuniqueness of the remainders obtained. This major difficulty in extending the idea of polynomial models of Fuhrmann to the rings of stable and causal stable rational functions is circumvented by choosing a remainder in a special form. A natural realization theory is thus developed for matrix fraction representations of transfer-function matrices over these rings. As an application of the new theory developed, linear matrix equations (QX+RY=T and QX+YR=T) over the rings of stable and causal stable rational functions are reduced to finite sets of linear equations over the base field.     

Answer to a conjecture of J.M. Cohen

The origin of Gelfand rings comes from [9] where the Jacobson topology and the weak topology are compared. The equivalence of these topologies defines a regular Banach algebra. One of the interests of these rings resides in the fact that we have an equivalence of categories between vector bundles over a compact manifold and finitely generated projective modules over C(M), the ring of continuous real functions on M [17].These rings have been studied by R. Bkouche (soft rings [3]) C.J. Mulvey (Gelfand rings [15]) and S. Teleman (harmonic rings [19]).Firstly we study these rings geometrically (by sheaves of modules (Theorem 2.5)) and then introduce the ?ech covering dimension of their maximal spectrums. This allows us to study the stable rank of such a ring A (Theorem 6.1), the nilpotence of the nilideal of K₀(A) - The Grothendieck group of the category of finitely generated projective A-modules - (Theorem 9.3), and an upper limit on the maximal number of generators of a finitely generated A-module as a function of the afore-mentioned dimension (Theorem 4.4).Moreover theorems of stability are established for the group K₀(A), depending on the stable rank (Theorems 8.1 and 8.2). They can be compared to those for vector bundles over a finite dimensional paracompact space [18].Thus there is an analogy between finitely generated projective modules over Gelfand rings and ?ech dimension, and finitely generated projective modules over noetherian rings and Krull dimension.     

Quaternion rings and octonion rings

In this paper, for rings R, we introduce complex rings ?(R), quaternion rings ?(R), and octonion rings O(R), which are extension rings of R; R ? ?(R) ? ?(R) ? O(R). Our main purpose of this paper is to show that if R is a Frobenius algebra, then these extension rings are Frobenius algebras and if R is a quasi-Frobenius ring, then ?(R) and ?(R) are quasi-Frobenius rings and, when Char(R) = 2, O(R) is also a quasi-Frobenius ring.     

Reversibility of Rings with Respect to the Jacobson Radical

Let R be a ring with identity and J(R) denote the Jacobson radical of R. A ring R is called J-reversible if for any a, \(b \in R\), \(ab = 0\) implies \(ba \in J(R)\). In this paper, we give some properties of J-reversible rings. We prove that some results of reversible rings can be extended to J-reversible rings for this general setting. We show that J-quasipolar rings, local rings, semicommutative rings, central reversible rings and weakly reversible rings are J-reversible. As an application it is shown that every J-clean ring is directly finite.     

Efficient algorithms for wavelength assignment on trees of rings

A fundamental problem in communication networks is wavelength assignment (WA): given a set of routing paths on a network, assign a wavelength to each path such that the paths with the same wavelength are edge-disjoint, using the minimum number of wavelengths. The WA problem is NP-hard for a tree of rings network which is well used in practice. In this paper, we give an efficient algorithm which solves the WA problem on a tree of rings with an arbitrary (node) degree using at most 3L wavelengths and achieves an approximation ratio of 2.75 asymptotically, where L is the maximum number of paths on any link in the network. The 3L upper bound is tight since there are instances of the WA problem that require 3L wavelengths even on a tree of rings with degree four. We also give a 3L and 2-approximation (resp. 2.5-approximation) algorithm for the WA problem on a tree of rings with degree at most six (resp. eight). Previous results include: 4L (resp. 3L) wavelengths for trees of rings with arbitrary degrees (resp. degree at most eight), and 2-approximation (resp. 2.5-approximation) algorithm for trees of rings with degree four (resp. six).     

Rings and semigroups with permutable zero products

We consider rings R, not necessarily with 1, for which there is a nontrivial permutation σ on n letters such that x₁?x_n=0 implies x_σ(1)?x_σ(n)=0 for all x₁,…,x_n∈R. We prove that this condition alone implies very strong permutability conditions for zero products with sufficiently many factors. To this end we study the infinite sequences of permutation groups P_n(R) consisting of those permutations σ on n letters for which the condition above is satisfied in R. We give the full characterization of such sequences both for rings and for semigroups with 0. This enables us to generalize some recent results by Cohn on reversible rings and by Lambek, Anderson and Camillo on rings and semigroups whose zero products commute. In particular, we prove that rings with permutable zero products satisfy the Köthe conjecture.     

Almost perfect commutative rings

Almost perfect commutative rings R are introduced (as an analogue of Bazzoni and Salce's almost perfect domains) for rings with divisors of zero: they are defined as orders in commutative perfect rings such that the factor rings $R/Rr$ are perfect rings (in the sense of Bass) for all non-zero-divisors$r\in R$. It is shown that an almost perfect ring is an extension of a T-nilpotent ideal by a subdirect product of a finite number of almost perfect domains. Noetherian almost perfect rings are exactly the one-dimensional Cohen–Macaulay rings. Several characterizations of almost perfect domains carry over practically without change to almost perfect rings. Examples of almost perfect rings with zero-divisors are abundant.     

Zero sets of polynomials in finite rings

In a field, of course, ann-th degree polynomial has at mostn zeros. Is there anything like this in rings? We find: ann-th degree polynomial in a finite ringA, ofr elements, that is not zero everywhere inA is non-zero at leastr/2 ⁿ times. This is sharp for alln, even in commutative rings; perhaps also in unitary rings, though examples are lacking beyond the cubic case. However, the best such result for commutative unitary rings (not determined pastn=3) is better; (2/2 ⁿ )r is proved, and the best coefficient is between that and its square root.     

Reflexive rings and their extensions

A right ideal I is reflexive if xRy ∈ I implies yRx ∈ I for x, y ∈ R. We shall call a ring R a reflexive ring if aRb = 0 implies bRa = 0 for a, b ∈ R. We study the properties of reflexive rings and related concepts. We first consider basic extensions of reflexive rings. For a reduced iedal I of a ring R, if R/I is reflexive, we show that R is reflexive. We next discuss the reflexivity of some kinds of polynomial rings. For a quasi-Armendariz ring R, it is proved that R is reflexive if and only if R[x] is reflexive if and only if R[x; x ^?1] is reflexive. For a right Ore ring R with Q its classical right quotient ring, we show that if R is a reflexive ring then Q is also reflexive. Moreover, we characterize weakly reflexive rings which is a weak form of reflexive rings and investigate its properties. Examples are given to show that weakly reflexive rings need not be semicommutative. It is shown that if R is a semicommutative ring, then R[x] is weakly reflexive.     

Almost slender rings and compact rings

Classical results concerning slenderness for commutative integral domains are generalized to commutative rings with zero divisors. This is done by extending the methods from the domain case and bringing them in connection with results on the linear topologies associated to non-discrete Hausdorff filtrations. In many cases a weakened notion “almost slenderness” of slenderness is appropriate for rings with zero divisors. Special results for countable rings are extended to rings said to be of “bounded type” (including countable rings, ‘small’ rings, and, for instance, rings that are countably generated as algebras over an Artinian ring).More precisely, for a ring R of bounded type it is proved that R is slender if R is reduced and has no simple ideals, or if R is Noetherian and has no simple ideals; moreover, R is almost slender if R is not perfect (in the sense of H. Bass). We use our methods to study various special classes of rings, for instance von Neumann regular rings and valuation rings. Among other results we show that the following two rings are slender: the ring of Puiseux series over a field and the von Neumann regular ring $k^{\mathbb{N}}/k^{(\mathbb{N})}$ over a von Neumann regular ring k.For a Noetherian ring R we prove that R is a finite product of local complete rings iff R satisfies one of several (equivalent) conditions of algebraic compactness. A 1-dimensional Noetherian ring is outside this ‘compact’ class precisely when it is almost slender. For the rings of classical algebraic geometry we prove that a localization of an algebra finitely generated over a field is either Artinian or almost slender. Finally, we show that a Noetherian ring R is a finite product of local complete rings with finite residue fields exactly when there exists a map of R-algebras $R^{\mathbb{N}}\to R$ vanishing on $R^{(\mathbb{N})}$.     

On Additive Maps and Commutativity in Rings

We study additive maps which are skew-commuting or skew-centralizing on appropriate subsets of a ring R; and we investigate commutativity in prime and semiprime rings admitting a nonzero derivation d such that [d(x),d(y)] = 0 for all x,y in some nonzero one-sided ideal. This paper has two main parts. The first, motivated by a recent result of Bre?ar [3] on triviality of skew-commuting additive maps on prime rings, is a study of additive maps which are skew-commuting or skew-centralizing on subsets of certain rings. The second continues a study, begun years ago by Herstein [7], of prime and semiprime rings R admitting a nonzero derivation d such that d(x)d(y) ? d(y)d(x) = 0 for all x, y in a suitably chosen subset of R.     

Generic noncommutative surfaces

We study a class of noncommutative surfaces, and their higher dimensional analogs, which come from generic subalgebras of twisted homogeneous coordinate rings of projective space. Such rings provide answers to several open questions in noncommutative projective geometry. Specifically, these rings R are the first known graded algebras over a field k which are noetherian but not strongly noetherian: in other words, R⊗_kB is not noetherian for some choice of commutative noetherian extension ring B. This answers a question of Artin, Small, and Zhang. The rings R are also maximal orders, but they do not satisfy all of the χ conditions of Artin and Zhang. In particular, they satisfy the χ₁ condition but not χ_i for i?2, answering a question of Stafford and Zhang and a question of Stafford and Van den Bergh. Finally, we show that the noncommutative scheme R-proj has finite global dimension.     

On the Behrens radical of matrix rings and polynomial rings

It is shown that the Behrens radical of a polynomial ring, in either commuting or non-commuting indeterminates, has the form of “polynomials over an ideal”. Moreover, in the case of non-commuting indeterminates, for a given coefficient ring, the ideal does not depend on the cardinality of the set of indeterminates. However, in contrast to the Brown-McCoy radical, it can happen that the polynomial ring R[X] in an infinite set X of commuting indeterminates over a ring R is Behrens radical while the polynomial ring R〈X〉 in an infinite set Y of non-commuting indeterminates over R is not Behrens radical. This is connected with the fact that the matrix rings over Behrens radical rings need not be Behrens radical. The class of Behrens radical rings, which is closed under taking matrix rings, is described.     

Primitive Idempotents of Schur Rings

In this paper, we explore the nature of central idempotents of Schur rings over finite groups. We introduce the concept of a lattice Schur ring and explore properties of these kinds of Schur rings. In particular, the primitive, central idempotents of lattice Schur rings are completely determined. For a general Schur ring S, S contains a maximal lattice Schur ring, whose central, primitive idempotents form a system of pairwise orthogonal, central idempotents in S. We show that if S is a Schur ring with rational coefficients over a cyclic group, then these idempotents are always primitive and are spanned by the normal subgroups contained in S. Furthermore, a Wedderburn decomposition of Schur rings over cyclic groups is given. Some examples of Schur rings over non-cyclic groups will also be explored.