Rings and Modules with Divisibility on Chains

An R-moduleM is defined to satisfy ACCd (resp. DCCd) on submodules if for every ascending (resp. descending) chain {Mi} of submodules of M, Mi = φi(Mi+1) (resp. Mi+1 = φi(Mi)) for some φi ∈ EndR(M) for i ≫ 0. We show that every nonzero submodule of an R-module with ACCd (resp. DCCd) on submodules contains a uniform submodule. As a consequence, a regular ring with ACCd (resp. DCCd) on right ideals has essential right socle. We also show that the endomorphism ring of a finitely generated self-projective module with ACCd (resp. DCCd) on submodules satisfy ACCd (resp. DCCd) on right ideals. Finally, any right self-injective ring with ACCd (resp. DCCd) on right ideals has finite right uniform dimension.