Some characterizations of Krull domains

In this paper, we define the v-finiteness for a length function L_v on the set of all v-ideals of an integral domain R and show that R is a Krull domain if and only if every proper integral v-ideal of R has v-finite length and L_v((AB)_v)=L_v(A)+L_v(B) for every pair of proper integral v-ideals A and B in R. We also give Euclidean-like characterizations of factorial, Krull, and π-domains. Finally we define the notion of quasi-∗-invertibility and show that if every proper prime t-ideal of an integral domain R is quasi-t-invertible, then R is a Krull domain.     

Power series extensions of half-factorial domains

An integral domain is said to be a half-factorial domain (HFD) if every non-zero element a that is not a unit may be factored into a finite product of irreducible elements, while any other such factorization of a has the same number of irreducible factors. While it is known that a power series extension of a factorial domain need not be factorial, the corresponding question for HFD has been open. In this paper we show that the answer is also negative. In the process we answer in the negative, for HFD, an open question of Samuel for factorial domains by showing that for certain quadratic domains R, and independent variables, Y and T, R[[Y]][[T]] is not HFD even when R[[Y]] is HFD. The proof hinges on Samuel’s theorem to the effect that a power series, in finitely many variables, over a regular factorial domain is factorial.     

Integral domains having a unique Kronecker function ring

We study the class of integrally closed domains having a unique Kronecker function ring, or equivalently, domains in which the completion (or b-operation) is the only e.a.b star operation of finite type. Such domains are a generalization of Prüfer domains and have fairly simple sets of valuation overrings. We give characterizations by studying valuation overrings and integral closure of finitely generated ideals. We provide new examples of such domains and show that for several well-known classes of integral domains the property of having a unique Kronecker function ring makes them fall into the class of Prüfer domains.     

Unique representation domains, II

Given a star operation ∗ of finite type, we call a domain R a ∗-unique representation domain (∗-URD) if each ∗-invertible ∗-ideal of R can be uniquely expressed as a ∗-product of pairwise ∗-comaximal ideals with prime radical. When ∗ is the t-operation we call the ∗-URD simply a URD. Any unique factorization domain is a URD. Generalizing and unifying results due to Zafrullah [M. Zafrullah, On unique representation domains, J. Nat. Sci. Math. 18 (1978) 19-29] and Brewer-Heinzer [J.W. Brewer, W.J. Heinzer, On decomposing ideals into products of comaximal ideals, Comm. Algebra 30 (2002) 5999-6010], we give conditions for a ∗-ideal to be a unique ∗-product of pairwise ∗-comaximal ideals with prime radical and characterize ∗-URD’s. We show that the class of URD’s includes rings of Krull type, the generalized Krull domains introduced by El Baghdadi and weakly Matlis domains whose t-spectrum is treed. We also study when the property of being a URD extends to some classes of overrings, such as polynomial extensions, rings of fractions and rings obtained by the D+XD_S[X] construction.     

Factorization of singular integer matrices

It is well known that a singular integer matrix can be factorized into a product of integer idempotent matrices. In this paper, we prove that every n  × n (n > 2) singular integer matrix can be written as a product of 3n + 1 integer idempotent matrices. This theorem has some application in the field of synthesizing VLSI arrays and systolic arrays.     

d-ideals,f d-ideals and prime ideals

Abstract

Let R be a commutative ring. An ideal I of R is called a d-ideal (f d-ideal) provided that for each a ∈ I (finite subset F of I) and b ∈ R, Ann(a) ? Ann(b) (Ann(F) ? Ann(b)) implies that b ∈ I. It is shown that, the class of z⁰-ideals (hence all sz⁰-ideals), maximal ideals in an Artinian or in a Kasch ring, annihilator ideals, and minimal prime ideals over a d-ideal are some distinguished classes of d-ideals. Furthermore, we introduce the class of f d-ideals as a subclass of d-ideals in a commutative ring R. In this regard, it is proved that the ring R is a classical ring with property (A) if and only if every maximal ideal of R is an f d-ideal. The necessary and sufficient condition for which every prime f d-ideal of a ring R being a maximal or a minimal prime ideal is given. Moreover, the rings for which their prime d-ideals are z⁰-ideals are characterized. Finally, we prove that every prime f d-ideal of a ring R is a minimal prime ideal if and only if for each a ∈ R there exists a finitely generated ideal , for some n ∈ ? such that Ann(a, I) = 0. As a consequence, every prime f d-ideal in a reduced ring R is a minimal prime ideal if and only if X= Min(R) is a compact space.     

Idempotent elements determined matrix algebras

Let M_n(R) be the algebra of all n×n matrices over a unital commutative ring R with 2 invertible, V be an R-module. It is shown in this article that, if a symmetric bilinear map {·,·} from M_n(R)×M_n(R) to V satisfies the condition that {u,u}={e,u} whenever u²=u, then there exists a linear map f from M_n(R) to V such that . Applying the main result we prove that an invertible linear transformation θ on M_n(R) preserves idempotent matrices if and only if it is a Jordan automorphism, and a linear transformation δ on M_n(R) is a Jordan derivation if and only if it is Jordan derivable at all idempotent points.     

Finite character of finitely stable domains

A commutative domain is finitely stable if every nonzero finitely generated ideal is stable, i.e. invertible over its endomorphism ring. A domain satisfies the local stability property provided that every locally stable ideal is stable.We prove that a finitely stable domain satisfies the local stability property if and only if it has finite character, that is every nonzero ideal is contained in at most finitely many maximal ideals. This result allows us to answer the open problem of whether every Clifford regular domain is of finite character.     

An extension of a theorem of Frobenius and Stickelberger to modules of projective dimension one over a factorial domain

Let R be a Cohen–Macaulay ring. A quasi-Gorenstein R-module is an R-module such that the grade of the module and the projective dimension of the module are equal and the canonical module of the module is isomorphic to the module itself. After discussing properties of finitely generated quasi-Gorenstein modules, it is shown that this definition allows for a characterization of diagonal matrices of maximal rank over a Cohen–Macaulay factorial domain R extending a theorem of Frobenius and Stickelberger to modules of projective dimension 1 over a commutative factorial Cohen–Macaulay domain.     

Neat rings

A ring is called clean if every element is the sum of a unit and an idempotent. Throughout the last 30 years several characterizations of commutative clean rings have been given. We have compiled a thorough list, including some new equivalences, in hopes that in the future there will be a better understanding of this interesting class of rings. One of the fundamental properties of clean rings is that every homomorphic image of a clean ring is clean. We define a neat ring to be one for which every proper homomorphic image is clean. In particular, the ring of integers, Z, and any nonlocal PID are examples neat rings which are not clean. We characterize neat Bézout domains using the group of divisibility. In particular, it is shown that a neat Bézout domain has stranded primes, that is, for every nonzero prime ideal the set of primes either containing or contained in the given prime forms a chain under set-theoretic inclusion.     

Valuations with preassigned proximity relations

We characterize the infinite upper triangular matrices (which we call formal proximity matrices) that can arise as proximity matrices associated with zero-dimensional valuations dominating regular noetherian local rings. In particular, for every regular noetherian local ring R of the appropriate dimension, we give a sufficient condition for such a formal proximity matrix to be the proximity matrix associated with a real rank one valuation dominating R. Furthermore, we prove that in the special case of rational function fields, each formal proximity matrix arises as the proximity matrix of a valuation whose value group is computable from the formal proximity matrix. We also give an example to show that this is false for more general fields. Finally in the case of characteristic zero, our constructions can be seen as a particular case of a structure theorem for zero-dimensional valuations dominating equicharacteristic regular noetherian local rings.     

Rational power series, sequential codes and periodicity of sequences

Let R be a commutative ring. A power series f∈R[[x]] with (eventually) periodic coefficients is rational. We show that the converse holds if and only if R is an integral extension over Z_m for some positive integer m. Let F be a field. We prove the equivalence between two versions of rationality in F[[x₁,…,x_n]]. We extend Kronecker’s criterion for rationality in F[[x]] to F[[x₁,…,x_n]]. We introduce the notion of sequential code which is a natural generalization of cyclic and even constacyclic codes over a (not necessarily finite) field. A truncation of a cyclic code over F is both left and right sequential (bisequential). We prove that the converse holds if and only if F is algebraic over F_p for some prime p. Finally, we show that all sequential codes are obtained by a simple and explicit construction.     

Local rings whose modules are almost serial

The present paper is a sequel to our previous work on almost uniserial rings and modules, which appeared in the Journal of Algebra in 2016; it studies rings over which every (left and right) module is almost serial. A module is almost uniserial if any two of its submodules are either comparable in inclusion or isomorphic. And a module is almost serial if it is a direct sum of almost uniserial modules. The results of the paper are inspired by a characterization of Artinian serial rings as rings having all left (or right) modules serial. We prove that if R is a local ring and all left R-modules are almost serial then R is an Artinian ring which is uniserial either on the left or on the right. We also produce a connection between local rings having all left and right modules almost serial, local balanced rings studied by Dlab and Ringel and local Köthe rings. Finally we prove Morita invariance of the almost serial property and list some consequences.     

Sharpening the triple diagonal form

Let R be a Euclidean domain with quotient field F of characteristic not equaling 2. Jacobi showed that every symmetric R-matrix is congruent over R to a matrix in triple diagonal form. Since it is generally not possible to fully diagonalize these matrices, it is of importance to gain as much control as possible of this triple diagonal form. This paper focuses on controlling the off-diagonal elements.     

Construction and nonnegativity of sign idempotent sign pattern matrices

In this paper, we modify Eschenbach’s algorithm for constructing sign idempotent sign pattern matrices so that it correctly constructs all of them. We find distinct classes of sign idempotent sign pattern matrices that are signature similar to an entrywise nonnegative sign pattern matrix. Additionally, if for a sign idempotent sign pattern matrix A there exists a signature matrix S such that SAS is nonnegative, we prove such S is unique up to multiplication by -1 if the signed digraph D(A) is not disconnected.     

Two criteria to check whether ideals are direct sums of cyclically presented modules

A famous theorem of commutative algebra due to I. M. Isaacs states that “if every prime ideal of R is principal, then every ideal of R is principal”. Therefore, a natural question of this sort is “whether the same is true if one weakens this condition and studies rings in which ideals are direct sums of cyclically presented modules?” The goal of this paper is to answer this question in the case R is a commutative local ring. We obtain an analogue of Isaacs's theorem. In fact, we give two criteria to check whether every ideal of a commutative local ring R is a direct sum of cyclically presented modules, it suffices to test only the prime ideals or structure of the maximal ideal of R. As a consequence, we obtain: if R is a commutative local ring such that every prime ideal of R is a direct sum of cyclically presented R-modules, then R is a Noetherian ring. Finally, we describe the ideal structure of commutative local rings in which every ideal of R is a direct sum of cyclically presented R-modules.     

Combining triple diagonal forms

Let R be a Euclidean domain with quotient field F of characteristic not equaling 2. Jacobi showed that every symmetric R-matrix is congruent over R to a matrix in triple diagonal form. Since it is generally not possible to fully diagonalize these matrices, it is of importance to gain as much control as possible of this triple diagonal form. Two different refinements have since been made to Jacobi’s triple diagonal form. This paper works toward combining these refinements.     

Euclidean modules

In this paper, we introduce the notion of Euclidean module and weakly Euclidean ring. We prove the main result that a commutative ring R is weakly Euclidean if and only if every cyclic R-module is Euclidean, and also if and only if End( _R M) is weakly Euclidean for each cyclic R-moduleM. In addition, some properties of torsion-free Euclidean modules are presented.     

Rings whose ideals are isomorphic to trace ideals

Let R be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of R is isomorphic to some trace ideal of R. This paper studies this question and gives several answers. In particular, a complete answer is given in the case where R is local: it is proved in this paper that every ideal of R is isomorphic to a trace ideal if and only if R is an artinian Gorenstein ring, or a 1‐dimensional hypersurface with multiplicity at most 2, or a unique factorization domain.