‎For the subclasses $mathcal{M}_1$ and $mathcal{M}_2$ of‎ ‎monomorphisms in a concrete category $mathcal{C}$‎, ‎if $mathcal‎{M}_2subseteq mathcal{M}_1$‎, ‎then $mathcal{M}_1$-injectivity‎ ‎implies $mathcal{M}_2$-injectivity‎. ‎The Baer type criterion is about‎ ‎the converse of this fact‎. ‎In this paper‎, ‎we apply injectivity to the classes of $C$-dense‎, ‎$C$-closed‎ ‎monomorphisms‎. ‎The con...

let $r$ be an arbitrary ring with identity and $m$ a right $r$-module with $s=$ end$_r(m)$. the module $m$ is called {it rickart} if for any $fin s$, $r_m(f)=se$ for some $e^2=ein s$. we prove that some results of principally projective rings and baer modules can be extended to rickart modules for this general settings.

The main purpose of this article is to offer some characterizations of $delta$-double derivations on rings and algebras. To reach this goal, we prove the following theorem:Let $n > 1$ be an integer and let $mathcal{R}$ be an $n!$-torsion free ring with the identity element $1$. Suppose that there exist two additive mappings $d,delta:Rto R$ such that $$d(x^n) =Sigma^n_{j=1} x^{n-j}d(x)x^{j-1}+Si...

In ring theory, the notion of annihilator is an important tool for studying the structures. Many characterizations and structure theorems can be derived by using this notion. On the other hand, certain classes of rings (e.g., Baer rings and Rickart rings) are defined by considering annihilators ideals. In the present work, we introduce a class of rings which is close to the class of Rickart rin...