Let R be any ring . This thesis is based on the works done on [5] and ]6], it is shown that modules satisfying ascending or descending chain conditions (resp . acc and dcc) on non-summand submodules belongs to some particular 0cm 0cm 0pt; TEXT-ALIGN: justify" For ? an infinite cardinal number and M a unital right module over a ring R , we show that every well ordered ascending ( resp . descending) chain of essential submodules of M has cardinality less than ? if and only if every well ordered ascending (respectively descending) chain of submodules of M/ soc (M) has cardinality less than ?. We use this to show that a CS module with an ?-chain condition on essential submodules is a direct sum of a module with that same chain condition on all submodules and a semisimple module . Thus a CS module with fewer than ? generators has an ?-chain condition on essential submodules if and only if it has that same ?-chain condition on all submodules . As an application in the case of a ? 0 -chain condition , we describe the endomorphism ring of a continuous module with ascending chain condition on essential submodules. Necessary and sufficient conditions are given for a module over a Dedekind domain to satisfy the ascending chain condition on n-generated submodules for every positive integer or on submodules with uniform dimension at most for every positive integer n. Given a positive integer n, we shall say that an R-module M satisfies nd-acc provided every ascending chain of submodules with uniform dimension at most n terminates . In addition , an R-module M satisfies pand-acc provided M satisfies nd-acc for every positive integer n . we show that , if M be a reduced torsion module over a DVR , Then M satisfies pand-acc . Also , it is proved that if R is a (commutative) Dedekind domain and M an R-module with torsion submodule T then every fnite dimensional submodule of M is Noetherian if and only if T does not contain any non-zero injective submodule and every countably generated torsion-free submodule of M is projective.