Divisibility on chains of submodules

An R-module M is said to satisfy ACC(d) (resp. DCCd) on submodules if for every ascending (resp. descending) chain {M-i} of submodules of M, for some for i >> 0. A nonzero module with ACC(d) or DCCd on submodules contains an essential submodule which is a direct sum of uniform submodules almost all noetherian. We show that if M is a finitely generated self-projective and self-injective R-module with ACC(d) or DCCd on submodules, then M is a finite direct sum of uniform submodules. It is shown that a regular right self-injective ring with ACC(d) or DCCd on right ideals must be semisimple artinian. We also prove that if M is a nonzero nonsingular module over a right noetherian ring and E(M)((N)) satisfies ACC(d) or DCCd on submodules, then M is semisimple. Next we consider some conditions for modules with ACC(d) (resp. DCCd) on submodules to satisfy ACC (resp. DCC) on some families of prime submodules. Finally, we show that a commutative ring with DCCd on ideals has dimension at most 1.