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 .

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- May 29th 2010, 09:33 AMxixi[SOLVED] the cardinal number of a finite local 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 .

- May 29th 2010, 03:24 PMNonCommAlg
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 - May 30th 2010, 04:55 AMxixiThanks