Baer modules over valuation domains

Summary A module B over a commutative domain R is said to be a Baer module if Ext _R ¹ (B, T)=0for all torsion R-modules T. The case in which R is an arbitrary valuation domain is investigated, and it is shown that in this case Baer modules are necessarily free. The method employed is totally different from Griffith's method for R=Z which breaks down for non-hereditary rings.This research was partially supported by NSF Grants DMS-8400451 and DMS-8500933.     

Weakly distributive modules. Applications to supplement submodules

In this paper, we define and study weakly distributive modules as a proper generalization of distributive modules.We prove that, weakly distributive supplemented modules are amply supplemented. In a weakly distributive supplemented module every submodule has a unique coclosure. This generalizes a result of Ganesan and Vanaja. We prove that π-projective duo modules, in particular commutative rings, are weakly distributive. Using this result we obtain that in a commutative ring supplements are unique. This generalizes a result of Camillo and Lima. We also prove that any weakly distributive ⊕-supplemented module is quasi-discrete.     

Note on residually finite rings

This paper is on the subject of residually finite (= RF) modules and rings introduced by Varadarajan [93] and [98/99]. Specifically there are several theorems that simplify proofs and generalize some results of Varadarajan, namely.

Theorem 1. An RF right R-module is finitely bedded (= has finite essential socle iff M is finite.

Corollay. If T is a right RF woth just finitely many simple ringht R-modules, them R is fimite.

Theorem 2. A commutative ring R is residually finite iff every local ring R_m at a maximal ideal m is finite.     

Near-Rings and Partitioned Groups

Over a commutative ring R with identity, free modules M with 2 distinguished submodules are studied. The category Rep₂R of such objects M have the obvious morphisms between them, which are homomorphisms between .R-modules preserving the distinguished submodules. The endo-morphisms for each M constitute a subalgebra End_RM of the algebra End_RM and the readability of λ-generated R-algebras A as End_RM is considered for cardinals λ. Despite the fact that 4 is the minimal number of distinguished submodules for realizing any algebra over a field il, we are able to prove a similar result in Rep₂R for many rings R including R = Z and algebras which are cotorsion-free. Several examples illustrate the boarder line of our main result. The main theorem is applied for constructing Butler groups in [11]     

On Factorization in Modules

In two articles, Anderson and Valdes-Leon generalized the theory of factorization in integral domains to commutative rings with zero divisors and to modules. Here we investigate some factorization properties in modules and state a result that relates factorization properties of an R-module, M, to the factorization properties of M as an (R/Ann(M))-module. Furthermore, we will investigate when a polynomial module, M[x], has the bounded factorization property, assuming that M has this property.     

Linear mappings preserving coincidence of factorized and boundary ranks of matrices over semirings

A problem of characterization of linear bijective mappings preserving coincidence of factorized and boundary ranks over a semiring is considered. A complete classification of those mappings over antinegative commutative rings with a unit and without divisors of zero is obtained.     

Decomposing torsion modules

E. Matlis proved that if R is an integral domain with quotient field Q and K is the R-module Q/R, then all torsion R-modules decompose into a direct sum of local submodules if and only if K decomposes into a direct sum of local submodules. Thus K is a test module to determine whether torsion modules decompose. We generalize this result to commutative rings. If R is a commutative ring and a torsion theory of R is given by a Gabriel topology , then form the ring of quotients R_ and let K be the cokernel of the canonical ring homomorphism from R to R_. In some special cases, every -torsion R-module decomposes into a direct sum of local submodules if and only if K decomposes. However, there is an example where this is not the case. The principal result is: given R,  and K, there is a related filter _K of ideals of R, which is a subset of , such that all _K-pretorsion R-modules decompose into a direct sum of local submodules if and only if K decomposes. The relationship between  and _K is investigated.     

Die Verallgemeinerung eines Satzes von Bourbaki und einige Anwendungen

We generalize a theorem of Bourbaki: Let R be a noetherian ring and M a finitely generated torsionfree R-module with rank r. Assume further M to be free for all ∈ Spec R with depth ? 1. Then there exists a free submodule F in M such that M/F is isomorphic to an ideal in R. There are some applications due to E.G.Evans,Jr. and M. Auslander, concerning the group K_o (R) resp. reflexive R-modules and - in case R is Gorenstein - R-modules of finite length.     

On Fully Idempotent Modules

A submodule N of a module M is idempotent if N = Hom(M, N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands. Counterexamples are given to show that this result is not true in general. It is shown that over commutative Noetherian rings, the fully idempotent modules are precisely the semisimple modules. We also show that the commutative rings over which every module is fully idempotent are exactly the semisimple rings. Idempotent submodules of free modules are characterized.     

The theory of radicals in lattice-ordered rings

Special classes of associative lattice-ordered rings are introduced which are analogous to V. A. Andrunakievich's special classes of rings. The appropriate special radicals for them are defined. It is shown that the special classes ofl-rings are: 1) the class of alll-primaryl-rings; 2) the class of alll-primaryl-rings without locally nilpotentl-ideals (it is shown that the correspondingl-ideal is a union of nil-l-ideals of the ring); 3) the class ofl-rings not containing strictly positive divisors of zero; 4) the class of subdirectly indecomposablel-rings withl-idempotent core.Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 639–648, December, 1968.     

Linear Recurring Arrays, Linear Systems and Multidimensional Cyclic Codes over Quasi-Frobenius Rings

This paper generalizes the duality between polynomial modules and their inverse systems (Macaulay), behaviors (Willems) or zero sets of arrays or multi-sequences from the known case of base fields to that of commutative quasi-Frobenius (QF) base rings or even to QF-modules over arbitrary commutative Artinian rings. The latter generalization was inspired by the work of Nechaev et al. who studied linear recurring arrays over QF-rings and modules. Such a duality can be and has been suggestively interpreted as a Nullstellensatz for polynomial ideals or modules. We also give an algorithmic characterization of principal systems. We use these results to define and characterize n-dimensional cyclic codes and their dual codes over QF rings for n>1. If the base ring is an Artinian principal ideal ring and hence QF, we give a sufficient condition on the codeword lengths so that each such code is generated by just one codeword. Our result is the n-dimensional extension of the results by Calderbank and Sloane, Kanwar and Lopez-Permouth, Z. X. Wan, and Norton and Salagean for n=1.     

Gorenstein Homological Dimension with Respect to a Semidualizing Module and a Generalization of a Theorem of Bass

Let C be a semidualizing module for a commutative ring R. In this paper, we study the resulting modules of finite G _C -projective dimension in Bass class, showing that they admit G _C -projective precover. Over local ring, we prove that dim _R (M) ≤ 𝒢? _C  ? id _R (M) for any nonzero finitely generated R-module M, which generalizes a result due to Bass.     

Characteristics of Linear Differential Operators Over Commutative Algebras

Three definitions for characteristics of linear differential operators in the category of modules over a commutative unitary algebra are given. These definitions are compared with each other and some basic fact concerning their properties are proved. It is shown that for algebras without zero divisors the characteristic ideal is involutive and is the support of the symbolic module.     

Generalized Divisors and Total Reflexivity

The notion of generalized divisors on schemes is introduced by Hartshorne. It is shown that there exists a bijection between the set of all generalized divisors on a scheme X and the set of all reflexive coherent 𝒪 _X -modules which are locally free of rank one at generic points. This bijection, corresponds Cartier divisors to the set of all locally free sheaves of rank one. Our aim in this article is to study the class of generalized divisors that maps to totally reflexive coherent 𝒪 _X -modules, under this correspondence. We investigate this class of divisors, that will be called Gorenstein divisors, both over schemes and also over commutative noetherian rings. We show that this class of divisors has usual properties and fits well in the hierarchy of divisors that already exists in the literature.     

HIGH ORDER KÄHLER MODULES OF NONCOMMUTATIVE RING EXTENSIONS

We construct the high order Kähler modules of noncommutative ring extensions B/A and show their fundamental properties. Our Kähler modules represent not only high order left derivations for one-sided modules but also high order central derivations for bimodules, which are usual derivations. This new viewpoint enables us to prove new results which were not known even though B is an algebra over a commutative ring A. Our results are the decomposition of Kähler modules by an idempotent element, exact sequences of Kähler modules, the Kähler modules of factor rings, and the relation to separable extensions. In particular, our exact sequences of high order Kähler modules were not known even though B is commutative.     

$f$-投射与$f$-内射模

Let R be a ring. A fight R-module M is called f-projective if Ext^1 （M, N） = 0 for any f-injective right R-module N. We prove that （F-proj,F-inj） is a complete cotorsion theory, where （F-proj （F-inj） denotes the class of all f-projective （f-injective） right R-modules. Semihereditary rings, von Neumann regular rings and coherent rings are characterized in terms of f-projective modules and f-injective modules.     

Commutative rings whose proper ideals are direct sums of uniform modules

An interesting result, obtaining by some theorems of Asano, Köthe and Warfield, states that: “for a commutative ring R, every module is a direct sum of uniform modules if and only if R is an Artinian principal ideal ring.” Moreover, it is observed that: “every ideal of a commutative ring R is a direct sum of uniform modules if and only if R is a finite direct product of uniform rings.” These results raise a natural question: “What is the structure of commutative rings whose all proper ideals are direct sums of uniform modules?” The goal of this paper is to answer this question. We prove that for a commutative ring R, every proper ideal is a direct sum of uniform modules, if and only if, R is a finite direct product of uniform rings or R is a local ring with the unique maximal ideal ? of the form ? = U⊕S, where U is a uniform module and S is a semisimple module. Furthermore, we determine the structure of commutative rings R for which every proper ideal is a direct sum of cyclic uniform modules (resp., cocyclic modules). Examples which delineate the structures are provided.     

Continuité des dérivations et des épimorphismes dans certaines algèbres de Banach

Extending results by R. V. Garimella ([8], [9]) and V. Runde ([14]), we give conditions for a commutative Banach algebraA without divisors of zero which force every derivation onA and every epimorphism from another Banach algebra ontoA to be continuous.     

On maximal commutative subrings of non-commutative rings

We observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exactly three (resp., four) maximal commutative subrings are characterized. If R has acc or dcc on its commutative subrings containing the center, whose intersection with the nontrivial summands is trivial, then R is Dedekind-finite. It is observed that every Artinian commutative ring R, is a finite intersection of some Artinian commutative subrings of a non-commutative ring, in each of which, R is a maximal subring. The intersection of maximal ideals of all the maximal commutative subrings in a non-commutative local ring R, is a maximal ideal in the center of R. A ring R with no nontrivial idempotents, is either a division ring or a right ue-ring (i.e., a ring with a unique proper essential right ideal) if and only if every maximal commutative subring of R is either a field or a ue-ring whose socle is the contraction of that of R. It is proved that a maximal commutative subring of a duo ue-ring with finite uniform dimension is a finite direct product of rings, all of which are fields, except possibly one, which is a local ring whose unique maximal ideal is of square zero. Analogues of Jordan-Hölder Theorem (resp., of the existence of the Loewy chain for Artinian modules) is proved for rings with acc and dcc (resp., with dcc) on commutative subrings containing the center. A semiprime ring R has only finitely many maximal commutative subrings if and only if R has a maximal commutative subring of finite index. Infinite prime rings have infinitely many maximal commutative subrings.     

Locally perfect commutative rings are those whose modules have maximal submodules

R denotes a commutative ring. After Bass[B], a ring R is perfect in case every module has a projective cover. A ring R is a max ring provided that every nonzero i2-module has a maximal submodule. Bass characterized perfect rings as semilocal rings with T-nilpotent Jacobson radical J, and showed they are max rings. Moreover, Bass proved that R is perfect iff R satisfies the dec on principal ideals. Using Bass' theorems, the Hamsher-Koifman ([H],[K]) characterization of max R (see (3) ?(4) below), and the characterization of max R by the author via subdirectly irreducible quasi-injective R-modules, we obtain.