The notion of simple-direct-injective modules which are a generalization injective unifies $C2$ and $C3$-modules. In the present paper, we introduce semisimple-direct-injective module gives unified viewpoint $C2$, $C3$, SSP properties modules. It is proved that ring $R$ Artinian serial with Jacobson radical square zero if only every right $R$-module has and, for any family simple $R$-modules $\...

In an earlier paper [KRS] we defined and investigated the properties of the näıve blowup of an integral projective scheme X at a single closed point. In this paper we extend those results to the case when one näıvely blows up X at any suitably generic zero-dimensional subscheme Z. The resulting algebra A has a number of curious properties; for example it is noetherian but never strongly noether...

let $r=oplus_{nin bbb n_0}r_n$ be a noetherian homogeneous ring with local base ring $(r_0,frak{m}_0)$, $m$ and $n$ two finitely generated graded $r$-modules. let $t$ be the least integer such that $h^t_{r_+}(m,n)$ is not minimax. we prove that $h^j_{frak{m}_0r}(h^t_{r_+}(m,n))$ is artinian for $j=0,1$. also, we show that if ${rm cd}(r_{+},m,n)=2$ and $tin bbb n_0$, then $h^t_{frak{m}_0r}(h^2_{...

In this paper we introduce the notion of classical quasi-primary submodules that generalizes the concept of classical primary submodules. Then, we investigate decomposition and minimal decomposition into classical quasi-primary submodules. In particular, existence and uniqueness of classical quasi-primary decompositions in finitely generated modules over Noetherian rings are proved. More...

In an earlier paper [KRS] we defined and investigated the properties of the na¨ıve blowup of an integral projective scheme X at a single closed point. In this paper we extend those results to the case when one na¨ıvely blows up X at any suitably generic zero-dimensional subscheme Z. The resulting algebra A has a number of curious properties; for example it is noetherian but never strongly noeth...