HOMOGENEOUS FUNCTIONS DETERMINED BY CYCLIC SUBMODULES

Abstract

For a unital module V over a commutative ring R, let C denote the collection of cyclic submodules. The ring ?_R(V;C) = {f ε End_R V |f(C) ?C, ?C ε_R (V;C) has been the object of several recent studies in which the structure of ?_R(V;C) is related to the triple (V, R,C). Here we introduce a new ring H_R(V;C) containing ?(V;C) and investigate its structure in terms of the parameters (V, R, C).     

Chain conditions in modules with krull dimension

Any ring with Krull dimension satisfies the ascending chain condition on semiprime ideals. This result does not hold more generally for modules. In particular if Ris the first Weyl algebra over a field of characteristic 0 then there are Artinian R-modules which do not satisfy the ascending chain condition on prime submodules. However, if Ris a ring which satisfies a polynomial identity then any R-module with Krull dimension satisfies the ascending chain condition on prime submodules, and, if Ris left Noethe-rian, also the ascending chain condition on semiprime submodules.     

Loewy and Artinian modules over commutative rings

Sunto Si dimostra che se M è un modulo Artiniano con zoccolo semplice su un anello commutativo R, allora End_R(M) è commutativo locale Noetheriano completo e che M, come End_R (M)-modulo, è l'inviluppo iniettivo dell'unico End_R(M)-modulo semplice. Si dimostra anche che ogni modulo di Loewy con invarianti di Loewy finiti è Artiniano e si siudia la successione degli invarianti di Loewy di un tale modulo.     

NOTES ON NILPOTENCY OF NIL SUBRINGS OF ENDOMORPHISM RINGS OF MODULES

Abstract

A family K of right R-modules is called a natural class if K is closed under submodules, direct sums, infective hulls, and isomorphic copies. The main result of this note is the following: Let K be a natural class on Mod-R and M ε K. If M satisfies a.c.c. (or d.c.c.) on the set of submodules {N ? M: M/N ε K}, then each nil subring of End(M_R ) is nilpotent.     

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.     

On rings with associated elements

A principal right ideal of a ring is called uniquely generated if any two elements of the ring that generate the same principal right ideal must be right associated (i.e., if for all a,b in a ring R, aR = bR implies a = bu for some unit u of R). In the present paper, we study “uniquely generated modules” as a module theoretic version of “uniquely generated ideals,” and we obtain a characterization of a unit-regular endomorphism ring of a module in terms of certain uniquely generated submodules of the module among some other results: End(M) is unit-regular if and only if End(M) is regular and all M-cyclic submodules of a right R-module M are uniquely generated. We also consider the questions of when an arbitrary element of a ring is associated to an element with a certain property. For example, we consider this question for the ring R[x;σ]∕(xⁿ⁺¹), where R is a strongly regular ring with an endomorphism σ be an endomorphism of R.     

Endomorphism rings and the category of finitely generated projective modules

Given a left R-module M, we study the connection between the (right and left) properties of its endomorphism ring S=End(_RM) and the properties of the category σ _f [M] of all submodules of finitely M-generated left R-modules.     

利用图的完形刻画wsr1条件

本文证明了：对于拟投射右R-模M,End_R（M）满足wsr1条件（即弱的stable range one条件）当且仅当M_R是T-投射模;wsr1条件是左右对称的;对于von Neumann正则环R,R满足wsr1条件当且仅当R为单侧幺正则环．     

拟内射模中的消去定理

当左拟内射模Ｍ的自同态环Ｅｎｄ_RＭ为一Ｄｅｃｋｉｎｄ有限环时．Ｍ的任何两个相互同构的子模的左相关补子横也同构。     

Systèmes récursifs et algèbre de hadamard de suites récurrentes linéaires sur des anneaux commutatifs.

Abstract

Let 𝒢 be a simple finite dimensional Lie algebra over the complex numbers and let 𝒢¯ = 𝒢₁ ⊕…⊕ 𝒢 _k be a regular semisimple subalgebra of 𝒢 with each 𝒢 _i being a simple algebra of type A or C. It is shown that the lattice of submodules of a generalized Verma 𝒢-module constructed by parabolic induction starting from a simple torsion free 𝒢¯-module is almost always isomorphic to the lattice of submodules of an associated module formed as a quotient of a classical Verma module by a sum of Verma submodules. In particular, it is shown that the Mathieu admissible Verma modules involved have maximal submodules which are the sum of Verma modules.     

On prime submodules and primary decomposition

We characterize prime submodules of R × R for a principal ideal domain R and investigate the primary decomposition of any submodule into primary submodules of R × R.     

𝔏-Rickart Modules

It is well known that the Rickart property of rings is not a left-right symmetric property. We extend the notion of the left Rickart property of rings to a general module theoretic setting and define 𝔏-Rickart modules. We study this notion for a right R-module M _R where R is any ring and obtain its basic properties. While it is known that the endomorphism ring of a Rickart module is a right Rickart ring, we show that the endomorphism ring of an 𝔏-Rickart module is not a left Rickart ring in general. If M _R is a finitely generated 𝔏-Rickart module, we prove that End _R (M) is a left Rickart ring. We prove that an 𝔏-Rickart module with no set of infinitely many nonzero orthogonal idempotents in its endomorphism ring is a Baer module. 𝔏-Rickart modules are shown to satisfy a certain kind of nonsingularity which we term “endo-nonsingularity.” Among other results, we prove that M is endo-nonsingular and End _R (M) is a left extending ring iff M is a Baer module and End _R (M) is left cononsingular.     

Endomorphism Rings of Regular Modules over Wild Hereditary Algebras

Let X be an indecomposable regular module over a connected wild hereditary path-algebra. The main result is a factorization property for maps in the radical of End _H (X) and an upper bound for its degree of nilpotency. The bound is sharp if X has elementary quasi-top. In this, case the socle of End _H (X) can also be characterized.     

广义拟内射模的自同态环

本文的目的,是推广[1]中定理1.22和[2]中命题1(1).我们得到:设R是环,且Q=End_R(M),其中M是广义拟内射模.那么有(1)J(Q)=Z(Q);(2)Q/J(Q)是Von Neumann正则环.     

The formal theory of Hopf algebras Part II: The case of Hopf algebras

Abstract

The category Hopf_R of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category Hopf_R is locally presentable, it is coreflective in the category of bialgebras over R, over every R-algebra there exists a cofree Hopf algebra. (2) If, in addition, R is absoluty flat, then Hopf_R is reflective in the category of bialgebras as well, and there exists a free Hopf algebra over every R-coalgebra. Similar results are obtained for relevant subcategories of Hopf_R. Moreover it is shown that, for every commutative unital ring R, the so-called “dual algebra functor” has a left adjoint and that, more generally, universal measuring coalgebras exist.     

Regularity and Substructures of Hom

ABSTRACT

We define “regular” for maps in a Hom group. This notion specializes to the well-known notions of (Von Neumann) regular in rings and modules. A map f ∈ Hom _R (A,M) is regular if and only if Ker(f) ?^⊕ A and Im(f) ?^⊕ M. There exists a unique maximal regular End(M)-End(A)-submodule in Hom _R (A,M). We study regularity in Hom _R (A ₁ ⊕ A ₂, M ₁ ⊕ M ₂). The existence of a regular function Hom _R (A,M) implies the existence of projective summands of Hom _R (A,M) _{End R (A)} and of _{End R ( M )} Hom _R (A,M). We consider regularity in endomorphism rings, and generalize a theorem of Ware-Zelmanowitz. We examine connections between the maximum regular bimodule and other substructures of Hom, mention two generalizations of regularity, and raise some questions.     

Central localization and Gelfand-Kirillov dimension

LetR be a factor ring of the enveloping algebra of a finite dimensional Lie algebra over a fieldk. If the centre ofR, Z, consists of non-zero divisors inR, the ringR _z obtained by localizing at the non-zero elements ofZ becomes a finitely generated algebra over the fieldK which arises as the field of fractions ofZ. The Gelfand-Kirillov dimension of anR-moduleM is denotedd(M). In this paper it is shown that ifR _Z ⊗ _R M ≠ 0 thend(M) ≧d(R _Z ⊗ _R M) + tr. deg _k Z, whered (R _z ⊗M) is the Gelfand-Kirillov dimension ofR _z ⊗M) viewed as anR _z -module andR _z is viewed as a finitely generatedK-algebra (not as ak-algebra). The result is primarily of a technical nature.     

Classification of Modules with Two Distinguished Pure Submodules and Bounded Quotients

In this paper, we consider modules over principal ideal domains R. The objects are free R- modules F with two distinguished pure submodules F ₀ and F₁ with F ₀ ∩ F₁ = 0 and bounded quotient F/(F ₀ ⊕ F ₁) and morphisms are the usual R-homomorphisms which preserve the distinguished submodules. This category is denoted by cRep₂.R and its objects, we say the cR₂-modules are denoted by F = (F, F₀, F ₁). The rank of a cR₂-module F is the rank of the free R-module F. We will show that cR₂ -@#@ modules are direct sums of indecomposable cR₂-modules of rank 1 or 2. The infinite series of indecomposable cR₂-modules is well-known and given explicitly after our Main Theorem 1.4. The result was first shown for cR₂modules of finite rank in Arnold and Dugas [4], then for countable rank, using heavy machinery due to Hill and Megibben [25] in Files and Göbel [20]. Our proof for arbitrary rank is based on [20] and illustrates the importance of Hill’s notion of an axiom-3 family of modules. The Main Theorem is applied to a classification of Butler groups with two critical types. ^{1 2}     

Abelian groups flat as modules over their endomorphism ring

Let R be a commutative ring with identity and let M be an R-module. We examine the situation where for each prime ideal ρof R the set of all ρ-prime submodules of M is finite. In case R is Noetherian and M is finitely generated, we prove that this condition is equivalent to there being a positive integer n such that for every prime ideal ρ of R, the number of ρ-prime submodules of Mis less than or equal to n. We further show that in this case, there is at most one ρ-prime submodule for all but finitely many prime ideals ρ of R.     

Reductive Modes

A mode is an idempotent and entropic algebra. We show that each variety R_m of m-step left reductive Ω-modes is the Mal'cev product (relative to modes) of R_kand R_m-k. The dual result holds for varieties R _n ¹ of n-step right reductive Ω-modes. The main result says that the join R_m V R _n ¹ is independent and coincides with the Mal'cev product R_m ∘ R _n ¹ . We also give an equational characterization of this variety, and discuss the structure of such modes. This revised version was published online in June 2006 with corrections to the Cover Date.