Theorem 1 (Fitting Decomposition Theorem).
Given , it is clear that and for any positive integer . Therefore, we have an ascending chain of submodules
and a descending chain of submodules
Both chains must be finite, since has finite length. Therefore, we can find a positive integer such that
If , then . Therefore, for some . Write . Applying the to the first term, we get , so it is in . The second term is clearly in . So
Corollary 1 (Fitting Lemma).
In the theorem above, is either nilpotent ( for some ) or an automorphism iff is indecomposable.
Suppose first that is indecomposable. Then either or . If , then the lemma is proved. Suppose . In the former case, any is the image of some under , so and therefore is onto. If , then , so . This means is an automorphism. In the latter case, for any , so is nilpotent.
Remark. Another way of stating Fitting Lemma is to say that is a local ring iff the finite-length module is indecomposable. The (unique) maximal ideal in consists of all nilpotent endomorphisms (and its complement consists of, of course, the automorphisms).
|Date of creation||2013-03-22 17:29:26|
|Last modified on||2013-03-22 17:29:26|
|Last modified by||CWoo (3771)|
|Synonym||Fitting decomposition theorem|
|Defines||Fitting’s decomposition theorem|