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We study Morita-equivalent version of the Zariski cancellation problem.
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This paper is an exposition of the so-called injective Morita contexts (in which the connecting bimodule morphisms are injective) and Morita $alpha$contexts (in which the connecting bimodules enjoy some local projectivity in the sense of Zimmermann-H
uisgen). Motivated by situations in which only one trace ideal is in action, or the compatibility between the bimodule morphisms is not needed, we introduce the notions of Morita semi-contexts and Morita data, and investigate them. Injective Morita data will be used (with the help of static and adstatic modules) to establish equivalences between some intersecting subcategories related to subcategories of modules that are localized or colocalized by trace ideals of a Morita datum. We end up with applications of Morita $alpha$-contexts to $ast$-modules and injective right wide Morita contexts.
In this paper, we study on the primeness and semiprimeness of a Morita context related to the rings and modules. Necessary and sufficient conditions are investigated for an ideal of a Morita context to be a prime ideal and a semiprime ideal. In particular, we determine the conditions under which a Morita context is prime and semiprime.
Let $H$ and $L$ be two Hopf algebras such that their comodule categories are monoidal equivalent. We prove that if $H$ is a twisted Calabi-Yau (CY) Hopf algebra, then $L$ is a twisted CY algebra when it is homologically smooth. Especially, if $H$ is a Noetherian twisted CY Hopf algebra and $L$ has finite global dimension, then $L$ is a twisted CY algebra.
We show that every subset of vertices of a directed graph E gives a Morita equivalence between a subalgebra and an ideal of the associated Leavitt path algebra. We use this observation to prove an algebraic version of a theorem of Crisp and Gow: certain subgraphs of E can be contracted to a new graph G such that the Leavitt path algebras of E and G are Morita equivalent. We provide examples to illustrate how desingularising a graph, and in- or out-delaying of a graph, all fit into this setting.
In the present paper we develop a small cancellation theory for associative algebras with a basis of invertible elements. Namely, we study quotients of a group algebra of a free group and introduce three axioms for the corresponding defining relations. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a vector space and establish the corresponding structure theorems. We also provide a revision of a concept of Gr{o}bner basis for our rings and establish a greedy algorithm for the Ideal Membership Problem.
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