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Totally reflexive modules with respect to a semidualizing bimodule

Zhen Zhang, Xiaosheng Zhu, Xiaoguang Yan (2013)

Czechoslovak Mathematical Journal

Let S and R be two associative rings, let S C R be a semidualizing ( S , R ) -bimodule. We introduce and investigate properties of the totally reflexive module with respect to S C R and we give a characterization of the class of the totally C R -reflexive modules over any ring R . Moreover, we show that the totally C R -reflexive module with finite projective dimension is exactly the finitely generated projective right R -module. We then study the relations between the class of totally reflexive modules and the Bass class...

Towards a theory of Bass numbers with application to Gorenstein algebras

Shiro Goto, Kenji Nishida (2002)

Colloquium Mathematicae

The notion of Gorenstein rings in the commutative ring theory is generalized to that of Noetherian algebras which are not necessarily commutative. We faithfully follow in the steps of the commutative case: Gorenstein algebras will be defined using the notion of Cousin complexes developed by R. Y. Sharp [Sh1]. One of the goals of the present paper is the characterization of Gorenstein algebras in terms of Bass numbers. The commutative theory of Bass numbers turns out to carry over with no extra changes....

Transfer of derived equivalences from subalgebras to endomorphism algebras II

Shengyong Pan, Jiahui Yu (2024)

Czechoslovak Mathematical Journal

We investigate derived equivalences between subalgebras of some Φ -Auslander-Yoneda algebras from a class of n -angles in weakly n -angulated categories. The derived equivalences are obtained by transferring subalgebras induced by n -angles to endomorphism algebras induced by approximation sequences. Then we extend our constructions in T. Brüstle, S. Y. Pan (2016) to n -angle cases. Finally, we give an explicit example to illustrate our result.

Triangulated categories of periodic complexes and orbit categories

Jian Liu (2023)

Czechoslovak Mathematical Journal

We investigate the triangulated hull of orbit categories of the perfect derived category and the bounded derived category of a ring concerning the power of the suspension functor. It turns out that the triangulated hull corresponds to the full subcategory of compact objects of certain triangulated categories of periodic complexes. This specializes to Stai and Zhao’s result on the finite dimensional algebra of finite global dimension. As the first application, if A , B are flat algebras over a commutative...

T-Rickart modules

S. Ebrahimi Atani, M. Khoramdel, S. Dolati Pish Hesari (2012)

Colloquium Mathematicae

We introduce the notions of T-Rickart and strongly T-Rickart modules. We provide several characterizations and investigate properties of each of these concepts. It is shown that R is right Σ-t-extending if and only if every R-module is T-Rickart. Also, every free R-module is T-Rickart if and only if R = Z ₂ ( R R ) ⊕ R ' , where R’ is a hereditary right R-module. Examples illustrating the results are presented.

Trisections of module categories

José A. de la Peña, Idun Reiten (2007)

Colloquium Mathematicae

Let A be a finite-dimensional algebra over a field k. We discuss the existence of trisections (mod₊ A,mod₀ A,mod₋ A) of the category of finitely generated modules mod A satisfying exactness, standardness, separation and adjustment conditions. Many important classes of algebras admit trisections. We describe a construction of algebras admitting a trisection of their module categories and, in special cases, we describe the structure of the components of the Auslander-Reiten quiver lying in mod₀ A.

Currently displaying 261 – 280 of 294