正规化扩张和摸

设环S是环R的正规化扩张.本文讨论了R—模与S—模两者间的若干相关性质.     

N-Crossed Extensions of Crossed Modules

Abstract

We obtain cohomological δ-functors and a five term exact sequence associated to an extension of crossed modules which yields Eilenberg-MacLane cohomology theory and the classic five term exact sequence associated to an extension of groups, when we consider the particular situation of groups.     

Central Extensions of Precrossed Modules

We classify the precrossed module central extensions using the second cohomology group of precrossed modules. We relate these central extensions to the relative central group extensions of Loday, and to other notions of centrality defined in general contexts. Finally we establish a Universal Coefficient Theorem for the (co)homology of precrossed modules, which we use to describe the precrossed module central extensions in terms of the generalized Galois theory developed by Janelidze.     

Generic Extensions of Prinjective Modules

Let be an algebraically closed field and let be a finite partially ordered set of finite prinjective type. We study generic extensions of prinjective modules over the incidence -algebra of . We prove that there exist generic extensions of prinjective -modules and describe properties of the monoid of generic extensions.Partially supported by Polish KBN Grant 5 P0 3A 015 21.     

广义Fitting模的扩张

In this paper, we study the closeness of strongly （∞）-hopfian properties under some constructions such as the ring of Morita context, direct products, triangular matrix, fraction ring etc. Also, we prove that if M[X] is strongly hopfian （resp. strongly co-hopfian） in R[X]-Mod, then M is strongly hopfian （resp. strongly co-hopfian） in R-Mod.     

关于模的最大有理扩张

ｉ）设Ｍ_ｉ为一族左Ｒ－模,记Ｍ_ｉ为Ｍ_ｉ的最大有理扩张,本文首先考察了∑⊕Ｍ_ｉ和∑⊕Ｍ_ｉ之间的关系,并证明了如对环Ｒ上任何一族模上述两者相等的充要条件是环Ｒ上每一个模都有理完全．ｉ）利用ｉ）研究了无零因子环上模的最大有理扩张,得到了一些结果．     

On Induced Modules Over Ore Extensions

Abstract

Let R be a ring and S = R[x;σ, δ] its Ore extension. For an R-module M _R we investigate the uniform dimension and associated primes of the induced S-module M ? _R  S.     

Gorenstein平坦模与Frobenius扩张

设R■A是环的Frobenius扩张,其中A是右凝聚环,M是任意左A-模.首先证明了_AM是Gorenstein平坦模当且仅当M作为左R-模也是Gorenstein平坦模.其次,证明了Nakayama和Tsuzuku关于平坦维数沿着Frobenius扩张的传递性定理的"Gorenstein版本":若_AM具有有限Gorenstein平坦维数,则Gfd_A(M)=Gfd_R(M).此外,证明了若R■S是可分Frobenius扩张,则任意A-模(不一定具有有限Gorenstein平坦维数),其Gorenstein平坦维数沿着该环扩张是不变的.     

Frobenius扩张上的$n$-Gorenstein投射模和维数

In this paper, we study n-Gorenstein projective modules over Frobenius extensions and n-Gorenstein projective dimensions over separable Frobenius extensions. Let R ■ A be a Frobenius extension of rings and M any left A-module. It is proved that M is an n-Gorenstein projective left A-module if and only if A ■_RM and Hom_R(A, M) are n-Gorenstein projective left A-modules if and only if M is an n-Gorenstein projective left R-module. Furthermore, when R ■ A is a separable Frobenius extension, n-Gorenstein projective dimensions are considered.     

Extending Tilting Modules to One-Point Extensions by Projectives

Let k be an algebraically closed field, B be a finite dimensional k-algebra and A be the one-point extension of B by the projective B-module P ₀. We compare the posets 𝒯 _A and 𝒯 _B of tilting A-modules and B-modules, respectively. We prove that the restriction and the extension functors define morphisms of posets r: 𝒯 _A  → 𝒯 _B and e: 𝒯 _B  → 𝒯 _A such that re = id. Moreover, e induces a full embedding of the quiver of 𝒯 _B into that of 𝒯 _A , whose image is closed under successors, and mapping distinct connected components of the first into distinct connected components of the second.     

Hopf Galois扩张与Hopf模结构定理

设Ｈ是域ｋ上任意的Ｈｏｐｆ代数。本文首先讨论了右Ｈ＿扩张Ａ／Ａ￣（ｃｏＨ）与Ｈｏｐｆ模范畴，给出了Ａ／Ａ￣（ｃｏＨ）为右Ｈ－Ｇａｌｏｉｓ扩张的充分必要条件和Ｈｏｐｆ模范畴满足结构定理的若干等价条件．然后我们讨论了不可约作用与除环的Ｇａｌｏｉｓ扩张．     

Study of the Modules over Rings and Their Extensions

Ukrainian Mathematical Journal - We study the problem of transfer of the properties of some types of modules under identity preserving ring homomorphisms. These studies seem to be overlooked in the...     

Characteristic Modules of Dual Extensions and Groebner Bases

Let C be a finite dimensional directed algebra over an algebraically closed field k and A=A(C) the dual extension of C. The characteristic modules of A are constructed explicitly for a class of directed algebras, which generalizes the results of Xi. Furthermore, it is shown that the characteristic modules of dual extensions of a certain class of directed algebras admit the left Groebner basis theory in the sense of E. L. Green.     

When Essential Extensions of Finitely Generated Modules are Finitely Generated

For certain classes 𝒞 of R-modules, including singular modules or modules with locally Krull dimensions, it is investigated when every module in 𝒞 with a finitely generated essential submodule is finitely generated. In case 𝒞 = Mod-R, this means E(M)/M is Noetherian for any finitely generated module M_R. Rings R with latter property are studied and shown that they form a class 𝒬 properly between the class of pure semisimple rings and the class of certain max rings. Duo rings in 𝒬 are precisely Artinian rings. If R is a quasi continuous ring in 𝒬 then R ? A ⊕ T where A is a semisimple Artinian ring and T ∈ 𝒬 with Z(T_T) ≤_ess T_T.     

Convexity Theories V: Extensions of Absolutely and Totally Convex Modules

The theories of absolutely and totally convex modules are commutative theories and have many properties in common with the theory of modules. Therefore a typical module theoretic notion, namely extensions, is investigated here in these theories. Because of the close connections with Banach spaces extensions of Banach spaces are considered, too.