Throughout the paper, all rings are associative rings with identity 1. The set of all idempotents of a ring R is denoted by E(R). Also, for a subset X ⊆ R, we denote the right [resp., left] annihilator of X by r(X) [resp., (X)]. We call a ring R a left p.p.-ring [3], in brevity, an l.p.p.-ring, if every principal left ideal of R, regarded as a left R-module, is projective. Dually, we may define...

If I is a (two-sided) ideal of ring R, we let $${\text {ann}}_l(I)=\{r\in R\mid rI=0\},$$ {ann}}_r(I)=\{r\in Ir=0\},$$ and {ann}}(I)={\text {ann}}_l(I)\cap {\text {ann}}_r(I)$$ be the left, right double annihilators. An said to an annihilator if $$I={\text {ann}}(J)$$ for some J (equivalently, {ann}}({\text {ann}}(I))=I$$ ). We study ideals Leavitt path algebras graph $$C^*$$ -algebras. Let $$L...

We state several conditions under which comultiplication and weak comultiplication modulesare cyclic and study strong comultiplication modules and comultiplication rings. In particular,we will show that every faithful weak comultiplication module having a maximal submoduleover a reduced ring with a finite indecomposable decomposition is cyclic. Also we show that if M is an strong comultiplicati...