Divisibility on chains of submodules

Abstract

ABSTRACTAn R-module M is said to satisfy ACCd (resp. DCCd) on submodules if for every ascending (resp. descending) chain {Mi} of submodules of M, (resp. ) for some for i≫0. A nonzero module with ACCd 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 ACCd 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 ACCd 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 ACCd or DCCd on submodules, then M is semisimple. Next we consider some conditions for modules with ACCd (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.