Radical of a module

Let R be a ring and M a left R-module. A submodule N of M is called maximal or cosimple if the quotient M/N is a simple module. The radical of the module M is the intersection of all maximal submodules of M, :\mathrm{rad}(M) = \bigcap\, \{N \mid N \mbox{ is a maximal submodule of } M\} Equivalently, :\mathrm{rad}(M) = \sum\, \{S \mid S \mbox{ is a superfluous submodule of } M\} These definitions have direct dual analogues for \mathrm{soc}(M). ==Properties==

• In addition to the fact that \mathrm{rad}(M) is the sum of superfluous submodules, in a Noetherian module, \mathrm{rad}(M) itself is a superfluous submodule. In fact, if M is finitely generated over a ring, then \mathrm{rad}(M) itself is a superfluous submodule. This is because any proper submodule of M is contained in a maximal submodule of M when M is finitely generated. • A ring for which \mathrm{rad}(M)=\{0\} for every right R-module M is called a right V-ring. • For any module M, \mathrm{rad}(M/\mathrm{rad}(M)) is zero. • M is a finitely generated module if and only if the cosocle M/\mathrm{rad}(M) is finitely generated and \mathrm{rad}(M) is a superfluous submodule of M. ==See also==