It is well-known that a ring R is semiperfect if and only if RR (or RR) is a supplemented module. Considering weak supplements instead of supplements we show that weakly supplemented modules M are semilocal (i.e., M/Rad(M) is semisimple) and that R is a semilocal ring if and only if RR (or RR) is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie d...

In this paper, we study the concept of $mathcal C$-reticulation for the category $mathcal C$ whose objects are lattice-valued maps. The relation between the free objects in $mathcal C$ and the $mathcal C$-reticulation of rings and modules is discussed. Also, a method to construct $mathcal C$-reticulation is presented, in the case where $mathcal C$ is equational. Some relations between the conce...

Endomorphism rings of modules appear as the center of a ring, as the fix ring of a ring with group action or as the subring of constants of a derivation. This note discusses the question whether certain ∗-prime modules have a prime endomorphism ring. Several conditions are presented that guarantee the primeness of the endomorphism ring. The contours of a possible example of a ∗-prime module who...

Abstract Twisted étale groupoid algebras have recently been studied in the algebraic setting by several authors connection with an abstract theory of Cartan pairs rings. In this paper we show that extensions ample groupoids correspond a precise manner to Boolean inverse semigroups. particular, discrete twists over certain abelian semigroups, and they are classified Lausch’s second cohomology gr...