Regular additively inverse semirings.
Regular Rings and Baer Rings.
Regularity of semigroup rings.
Regularity of the four dimensional Sklyanin algebra
Remarks on strongly regular rings
Representation of algebraic distributive lattices with ℵ1 compact elements as ideal lattices of regular rings.
We prove the following result: Theorem. Every algebraic distributive lattice D with at most ℵ1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R.(By earlier results of the author, the ℵ1 bound is optimal.) Therefore, D is also isomorphic to the congruence lattice of a sectionally complemented modular lattice L, namely, the principal right ideal lattice of R. Furthermore, if the largest element of D is compact, then one can assume that R is unital, respectively,...
Restricted Regular Rings.
Rings consisting entirely of certain elements
We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; ; where is a Boolean ring; local ring with nil Jacobson radical; or ; or the ring of a Morita context with zero pairings where the underlying rings are or .
Rings decomposed into direct sums of J-rings and nil rings.
Rings generalized by tripotents and nilpotents
We present new characterizations of the rings for which every element is a sum of two tripotents and a nilpotent that commute. These extend the results of Z. L. Ying, M. T. Koşan, Y. Zhou (2016) and Y. Zhou (2018).
Rings which are semilattices of archimedean semigroups.
Rings whose Elements are Quasi-Regular or Regular
-pure submodules.
-weakly regular group rings
In this note we obtain a necessary and sufficient condition for a ring to be -weakly regular (i) When is a ring with identity and without divisors of zero (ii) When is a ring without divisors of zero. Further it is proved in a -weakly regular ring with identity and without units every element is a zero divisor.
Self-injective Von Neumann regular subrings and a theorem of Pere Menal.
This paper owes its origins to Pere Menal and his work on Von Neumann Regular (= VNR) rings, especially his work listed in the bibliography on when the tensor product K = A ⊗K B of two algebras over a field k are right self-injective (= SI) or VNR. Pere showed that then A and B both enjoy the same property, SI or VNR, and furthermore that either A and B are algebraic algebras over k (see [M]). This is connected with a lemma in the proof of the Hilbert Nullstellensatz, namely a finite ring extension...
Semiprime SF-rings whose essential left ideals are two-sided.
Semirings embedded in a completely regular semiring
Recently, we have shown that a semiring is completely regular if and only if is a union of skew-rings. In this paper we show that a semiring satisfying can be embedded in a completely regular semiring if and only if is additive separative.
Simple Regular Rings and Rank Functions.
Sobre anillos de polinomios que son anillos de Bézout.
Solutions of minus partial ordering equations over von Neumann regular rings
In this paper, we mainly derive the general solutions of two systems of minus partial ordering equations over von Neumann regular rings. Meanwhile, some special cases are correspondingly presented. As applications, we give some necessary and sufficient conditions for the existence of solutions. It can be seen that some known results can be regarded as the special cases of this paper.