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مضامين الموريتا الحقنية (مراجعة)
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هذا الورق هو تفسير لمحتويات موريتا المدخلية (التي تتضمن الأشكال الثنائية المربطة المدخلية) ومحتويات موريتا ألفا (التي تتمتع ببعض المشروعية المحلية في إطار ما يقال له زيمرمان-هويسجن). مشجعين بالأوضاع التي تتضمن وجود مثال واحد فقط من المثال المشارك، أو عدم الحاجة إلى التوافق بين الأشكال الثنائية المربطة، نقدم في هذا الورق المفاهيم الجديدة لمحتويات موريتا النصفية وبيانات موريتا، ونحاول التحقيق فيها. سيتم استخدام بيانات موريتا المدخلية (مع المساعدة من الوحدات الثابتة والأداستاتية) لإقامة التعادل بين بعض التصنيفات المتداخلة المتعلقة بتصنيفات الوحدات المحلولة أو المحلولة بواسطة مثال مشارك من بيانات موريتا. ننتهي بتطبيقات محتويات موريتا ألفا على الوحدات الأست ومحتويات موريتا المدخلية اليمنى الواسعة.
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-Huisgen). 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.
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Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no non-zero injective semimodules (e.g. the semiring of non-negative integers). In this paper, we study some of the basic properties of the so called e-injective semimodules introduced by the first author using a new notion of exact sequences of semimodules. We clarify the relationships between the injective semimodules, the e-injective semimodule, and the i-injective semimodules through several implications, examples and counter examples. Moreover, we provide partial results for the so called Embedding Problem (of semimodules in injective semimodules).
We study Morita-equivalent version of the Zariski cancellation problem.
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.
It is proved that localizations of injective $R$-modules of finite Goldie dimension are injective if $R$ is an arithmetical ring satisfying the following condition: for every maximal ideal $P$, $R_P$ is either coherent or not semicoherent. If, in addition, each finitely generated $R$-module has finite Goldie dimension, then localizations of finitely injective $R$-modules are finitely injective too. Moreover, if $R$ is a Prufer domain of finite character, localizations of injective $R$-modules are injective.
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.
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