If is a finite local commutative ring having residue field(= where is the unique maximal ideal of ) consisting of elements ( prime integer) then prove that the cardinal number of is i.e. where denotes the length of the (Artinian) ring .
If is a finite local commutative ring having residue field(= where is the unique maximal ideal of ) consisting of elements ( prime integer) then prove that the cardinal number of is i.e. where denotes the length of the (Artinian) ring .
every simple module over a commutative ring is isomorphic with where is some maximal ideal of now your ring has only one maximal ideal and so if is a simple module, then
In particular now let be a composition series for then for all because every
is a simple module. therefore