hollow matrix rings
Suppose that are both rings. The hollow matrix ring of is the ring of matrices:
It is easy to check that this forms a ring under the usual matrix addition and multiplication. This definition is slightly simplified from the obvious higher dimensional examples and the transpose of these matrices will also qualify as a hollow matrix ring.
2 Left/Right Artinian and Noetherian
We specialize to an example with the fields and , though the same argument can be made in much more general settings.
Let be a left ideal of and suppose that for some and .
Suppose that . Hence, for each and so for all . In particular, . So in all cases it follows that . So now we take and assume that does not contain any with . By observing matrix multiplication it follows that is now a left -vector space, and so any chain of left -modules is a chain of subspaces. As , it follows that such chains are finite.
Hence, there can be no infinite descending chain of distinct left ideals and so is left Artinian and Noetherian. ∎
is not right Artinian nor right Noetherian.
Using (the usual ), or any other transcendental number, we define
Since properly contains for all , it follows that is an infinite proper ascending and descending chain of right ideals. Therefore, is neither right Artinian nor right Noetherian. ∎
If is a ring with an anti-isomorphism, then the set of left ideals is mapped to the set of right ideals, bijectively and order preserving. This is not possible with . ∎
|Title||hollow matrix rings|
|Date of creation||2013-03-22 17:42:21|
|Last modified on||2013-03-22 17:42:21|
|Last modified by||Algeboy (12884)|