Semisimple module In abstract algebra , a module is said to be semisimple if it is the sum of simple submodules . Here , the base ring is a ring with unity , though not necessarily commutative . A semisimple module may be characterised as being ( isomorphic to ) a direct sum of simple modules . Properties Every submodule of a semisimple module is a direct summand . If M is semisimple and N is a submodule , then N and M / N are also semisimple . If each M_i is a semisimple module , then so is \oplus_i M_i . A module M is finitely generated and semisimple if and only if it is Artinian and its radical is zero . Semisimple Rings A ring is said to be ( left ) - semisimple if it is semisimple as a left module over itself . Surprisingly , a left-semisimple ring is also right-semisimple and vice versa . Hence , one often drops the left/right quantifier altogether and simply speaks of semisimple rings . Semisimple rings are of particular interest to algebraists . For example , if the base ring R is semisimple , then all R -modules would automatically be semisimple . Furthermore , every simple ( left ) R -module is isomorphic to a minimal left ideal of R . Semisimple rings are also small ( they 're both Artinian and Noetherian ) . From the above properties , a ring is semisimple if and only if it is Artinian and its radical is zero . Semisimple vs Simple Rings One should beware that despite the terminology , not all simple rings are semisimple . The problem is that the ring may be `` too big '' , and possibly not ( left/right ) Artinian . In fact , if R is a simple ring with a minimal left/right ideal , then R is semisimple . Examples If k is a field and G is a finite group of order n , then the group ring k [ G ] is semisimple if and only if the characteristic of k does not divide n . This is an important result in group representation theory . By Wedderburn 's structure theorem , a ring R is semisimple if and only if it is ( isomorphic to ) M_n ( D_1 ) \times M_n ( D_2 ) \times \dots \times M_n ( D_r ) , where each D_i is a division ring and M_n ( D ) is the ring of n -by- n matrices with entries in D . See also Socle Further reading R.S. Pierce . Associative Algebras . Graduate Texts in Mathematics vol 88 . T.Y. Lam . A First Course in Non-commutative Rings . Graduate Texts in Mathematics vol 131 . Categories : Module theory 