dense ring of linear transformations
Let be a division ring and a vector space over . Let be a subring of the ring of endomorphisms (linear transformations) of . Then is said to be a dense ring of linear transformations (over ) if we are given
any positive integer ,
any set of linearly independent vectors in , and
any set of arbitrary vectors in ,
then there exists an element such that
Note that the linear independence of the ’s is essential in insuring the existence of a linear transformation . Otherwise, suppose where . Pick ’s so that they are linearly independent. Then , contradicting the linear independence of the ’s.
First, assume that is a dense ring of linear transformations of . Recall that the compact-open topology on has subbasis of the form , where is open and is compact in . Since is discrete, is finite. Now, pick a point and let
be a neighborhood of , some index set. Then for some , . This means that for all . Since each is finite, so is . After some re-indexing, let be a maximal linearly independent subset of . Set , . By assumption, there is an such that , for all . For any , is a linear combination of the ’s: , . Then for some . This shows that and we have .
Conversely, assume that the ring is a dense subset of the space . Let be linearly independent, and be arbitrary vectors in . Let be the subspace spanned by the ’s. Because the ’s are linearly independent, there exists a linear transformation such that and for . Let and . Then the ’s are compact and the ’s are open in the discrete space . Clearly for each . So lies in the neighborhood . Since is dense in , there is an . This implies that for all . ∎
If is finite dimensional over , then any dense ring of linear transformations . This can be easily observed by using the second half of the proof above. Take a basis of and any set of vectors in . Let be the linear transformation that maps to . The above proof shows that there is an such that agrees with on the basis elements. But then they must agree on all of as a result, which is precisely the statement that .
It can be shown that a ring is a primitive ring iff it is isomorphic to a dense ring of linear transformations of a vector space over a division ring. This is known as the . It is a generalization of the special case of the Wedderburn-Artin Theorem when the ring in question is a simple Artinian ring. In the general case, the finite chain condition is dropped.
|Title||dense ring of linear transformations|
|Date of creation||2013-03-22 15:38:46|
|Last modified on||2013-03-22 15:38:46|
|Last modified by||CWoo (3771)|
|Defines||Jacobson Density Theorem|