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Semisimple module finite length

Noetherian module implies finite direct sum of indecomposables?A module decomposition problemModule and Noetherian/Artinian RingsQuotients of finitely generated module are direct summands implies module is semisimpleEquivalence of SemisimplicityShow that if $M$ is a semisimple artinian module then $M$ is finitely generated.Book for Module TheoryModule of finite length $implies$ finite direct sum of indecomposable modulesFinitely generated semisimple modulesA module $M$ whose submodules and factor modules are semisimple but not semisimple itself

0

$begingroup$

'm new in module theory and try to prove this theorem:

If $M$ is semisimple, the following statement is equivalent:

(1) $M$ is finitely generated.

(2) $M$ has finite length.

(3) $M$ is Noetherian and Artinian.

With definition, a finite length module $M$ is a module which isomorphic with a direct sum of finite simple submodules, it's easy to prove (2) to (1) and (3) to (2). Is there any proof for the other?

modules

share|cite|improve this question

edited Mar 12 at 11:00

Bernard

123k741116

asked Mar 12 at 10:25

HoàngHoàng

385

$endgroup$

add a comment | 

0

$begingroup$

'm new in module theory and try to prove this theorem:

If $M$ is semisimple, the following statement is equivalent:

(1) $M$ is finitely generated.

(2) $M$ has finite length.

(3) $M$ is Noetherian and Artinian.

With definition, a finite length module $M$ is a module which isomorphic with a direct sum of finite simple submodules, it's easy to prove (2) to (1) and (3) to (2). Is there any proof for the other?

modules

share|cite|improve this question

edited Mar 12 at 11:00

Bernard

123k741116

asked Mar 12 at 10:25

HoàngHoàng

385

$endgroup$

add a comment | 

$begingroup$

'm new in module theory and try to prove this theorem:

If $M$ is semisimple, the following statement is equivalent:

(1) $M$ is finitely generated.

(2) $M$ has finite length.

(3) $M$ is Noetherian and Artinian.

With definition, a finite length module $M$ is a module which isomorphic with a direct sum of finite simple submodules, it's easy to prove (2) to (1) and (3) to (2). Is there any proof for the other?

modules

share|cite|improve this question

edited Mar 12 at 11:00

Bernard

123k741116

asked Mar 12 at 10:25

HoàngHoàng

385

$endgroup$

share|cite|improve this question

edited Mar 12 at 11:00

Bernard

123k741116

asked Mar 12 at 10:25

HoàngHoàng

385

share|cite|improve this question

edited Mar 12 at 11:00

Bernard

123k741116

asked Mar 12 at 10:25

HoàngHoàng

385

edited Mar 12 at 11:00

Bernard

123k741116

edited Mar 12 at 11:00

Bernard

123k741116

asked Mar 12 at 10:25

HoàngHoàng

385

asked Mar 12 at 10:25

HoàngHoàng

385