Let R be an associative ring with identity. An element x 2 R is called Z G -regular (resp. strongly ZG -regular) if there exist g ÎG, nÎZ and rÎR such that xng=xngrxng (resp. xng=x (n+1) g).A ringR is called Z G -regular (resp. strongly Z G -regular) if every element of R is Z G -regular (resp.stronglyZ G -regular). In this paper, we characterize Z G -regular (resp. strongly Z G -regular) rings.Furthermore, this paper includes a brief discussion of Z G -regularity in group rings.

Let A be a unital R-algebra and M be a unital A- bimodule. It is shown that every Jordan derivation of the trivial extension of A by M, under some conditions, is the sum of a derivation and an antiderivation.

The aim of this paper is to study the (a; g) -prolongation of central extensions. We obtain the obstruction theory for (a; g) -prolongations and classify (a; g) -prolongations thanks to low-dimensional cohomology groups of groups.

We introduce the notion of nilpotent p.p. rings, and prove that the nilpotent p.p. condition is preserved over polynomial rings and skew polynomial rings.

A specialization semilattice is a join semilattice together with a coarser preorder ⊑,satisfying an appropriate compatibility condition. If 𝑋,is a topological space, then (P(𝑋, ), ∪, , ⊑, ) is a specialization semilattice, where 𝑥,⊑,𝑦,if 𝑥,⊆,𝐾, 𝑦, , for 𝑥, , 𝑦,⊆,𝑋, , and 𝐾,is closure. Specialization semilattices and posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology. In a former work we showed that every specialization semilattice can be embedded into the specialization semilattice associated to a topological space as above. Here we describe the universal embedding of a specialization semilattice into an additive closure semilattice.

We introduce a new formal logic, called equality propositional logic. It has two basic connectives, ^ (conjunction) and ≡ (equivalence). Moreover, the ⇒ (implication) connective can be derived as A ⇒ B: = (A ^ B) ≡ A. We formulate the equality propositional logic and demonstrate that the resulting logic has reasonable properties such as Modus Ponens(MP) rule, Hypothetical Syllogism(HS) rule and completeness, etc. Especially, we provide two ways to prove the completeness of this logic system. We also introduce two extensions of equality propositional logic. The first one is involutive equality propositional logic, which is equality propositional logic with double negation. The second one adds prelinearity which is rich enough to enjoy the strong completeness property. Finally, we introduce additional connective Δ (delta) in equality propositional logic and demonstrate that the resulting logic holds soundness and completeness.