Rings and Modules with Divisibility on Chains

n $R$-module $M$ is defined to satisfy $\rm{ACC_{d}}$ (resp. $\rm{DCC_{d}}$) on submodules if for every ascending (resp. descending) chain $\{M_{i}\}$ of submodules of $M$, $M_{i}=\varphi_{i}(M_{i+1})$ (resp. $M_{i+1}=\varphi_{i}(M_{i})$) for some $\varphi_{i}\in{\mathrm{End}_{R}(M)}$ for $i\gg0$. We show that every nonzero submodule of an $R$-module with $\rm{ACC_{d}}$ (resp. $\rm{DCC_{d}}$) on submodules contains a uniform submodule. As a consequence, a regular ring with $\rm{ACC_{d}}$ (resp. $\rm{DCC_{d}}$) on right ideals has essential right socle. We also show that the endomorphism ring of a finitely generated self-projective module with $\rm{ACC_{d}}$ (resp. $\rm{DCC_{d}}$) on submodules satisfy $\rm{ACC_{d}}$ (resp. $\rm{DCC_{d}}$) on right ideals. Finally, any right self-injective ring with $\rm{ACC_{d}}$ (resp. $\rm{DCC_{d}}$) on right ideals has finite right uniform dimension.