A survey of s-unital and locally unital rings

@article{Nystedt2018ASO,
  title={A survey of s-unital and locally unital rings},
  author={Patrik Nystedt},
  journal={arXiv: Rings and Algebras},
  year={2018}
}
We gather some classical results and examples that show strict inclusion between the families of unital rings, rings with enough idempotents, rings with sets of local units, locally unital rings, s-unital rings and idempotent rings. 
Prime group graded rings with applications to partial crossed products and Leavitt path algebras
In this article we generalize a classical result by Passman on primeness of unital strongly group graded rings to the class of nearly epsilon-strongly group graded rings which are not necessarilyExpand
The graded structure of algebraic Cuntz-Pimsner rings
The algebraic Cuntz-Pimsner rings are naturally Z-graded rings that generalize both Leavitt path algebras and unperforated Z-graded Steinberg algebras. We classify strongly, epsilon-strongly andExpand
Hochschild (Co)homology and Derived Categories
These are slightly expanded notes of lectures given in April 2019 at the Isfahan school and conference on representations of algebras. We recall the formalism of derived categories and functors andExpand
The graded structure of algebraic Cuntz-Pimsner rings
The algebraic Cuntz-Pimsner rings are naturally $\mathbb{Z}$-graded rings that generalize both Leavitt path algebras and unperforated $\mathbb{Z}$-graded Steinberg algebras. We classify strongly,Expand
A characterization of graded von Neumann regular rings with applications to Leavitt path algebras
We provide a characterization of graded von Neumann regular rings involving the recently introduced class of nearly epsilon-strongly graded rings. As our main application, we generalize Hazrat'sExpand
GRADED MODULES OVER OBJECT-UNITAL GROUPOID GRADED RINGS
In a previous article (see [6]), we introduced and analyzed ring-theoretic properties of object unital G-graded rings R, where G is a groupoid. In the present article, we analyze the category G-R-modExpand
Object-unital groupoid graded modules.
Given a groupoid $\mathcal{G}$ and an associative but not necessarily unital ring $R$, we introduce the notion of object unital graded ring and construct the category of object unital graded modulesExpand
Induced quotient group gradings of epsilon-strongly graded rings
Let $G$ be a group and let $S=\bigoplus_{g \in G} S_g$ be a $G$-graded ring. Given a normal subgroup $N$ of $G$, there is a naturally induced $G/N$-grading of $S$. It is well-known that if $S$ isExpand
Object-unital groupoid graded rings, crossed products and separability
Abstract We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal)Expand

References

SHOWING 1-10 OF 16 REFERENCES
A NOTE ON THE COMMUTATIVITY OF RINGS
Sufficient conditions for commutativity of rings are proved. They generalize or are related to certain old results due to I. N. Herstein and others, see [1] and [5].
Morita equivalence for rings withot identity
In the paper [1] Abrams made a first step in extending the theory of Morita equivalence to rings without identity. He considered rings in which a set of commuting idempotents is given such that everyExpand
MORITA EQUIVALENCE FOR RINGS WITHOUT IDENTITY
In the paper [1] Abrams made a first step in extending the theory of Morita equivalence to rings without identity. He considered rings in which a set of commuting idempotents is given such that everyExpand
Separable Groupoid Rings
We show that groupoid rings are separable over their ring of coefficients if and only if the groupoid is finite and the orders of the associated principal groups are invertible in the ring ofExpand
On rings whose left modules are direct sums of finitely generated modules
The relationship between rings of finite module type and rings whose left modules have decompositions that complement direct summands is examined by proving that the latter are precisely the rings ofExpand
Leavitt Path Algebras
The central concept of this thesis is that of Leavitt path algebras, a notion introduced by both Abrams and Aranda Pino in [AA1] and Ara, Moreno and Pardo in [AMP] in 2004. The idea of using a fieldExpand
Exercises In Classical Ring Theory
TLDR
This book provides a comprehensive set of problems and solutions in ring theory that will serve not only as a teaching aid to instructors using that book, but also for students, who will see how ring theory theorems are applied to solving ring-theoretic problems and how good proofs are written. Expand
Morita equivalence for rings with local units
Exercises in classical ring theory, 2nd edition, Springer-Verlag
  • New York,
  • 2003
The category of s-unital modules
...
1
2
...