example of a right noetherian ring that is not left noetherian
This example, due to Lance Small, is briefly described in Noncommutative Rings, by I. N. Herstein, published by the Mathematical Association of America, 1968.
It is relatively straightforward to show that is not left noetherian. For each natural number , let
Verify that each is a left ideal in and that .
Let be a right ideal in . We show that is finitely generated by checking all possible cases. In the first case, we assume that every matrix in has a zero in its upper left entry. In the second case, we assume that there is some matrix in that has a nonzero upper left entry. The second case splits into two subcases: either every matrix in has a zero in its lower right entry or some matrix in has a nonzero lower right entry.
CASE 1: Suppose that for all matrices in , the upper left entry is zero. Then every element of has the form
Note that for any and any , we have since
and is a right ideal in . So looks like a rational vector space.
Now, an arbitrary element in corresponds to the vector in and for some . Thus
and it follows that is finitely generated by the set as a right ideal in .
CASE 2: Suppose that some matrix in has a nonzero upper left entry. Then there is a least positive integer occurring as the upper left entry of a matrix in . It follows that every element of can be put into the form
By definition of , there is a matrix of the form in . Since is a right ideal in and since it follows that is in . Now break off into two subcases.
case 2.1: Suppose that every matrix in has a zero in its lower right entry. Then an arbitrary element of has the form
Note that . Hence, generates as a right ideal in .
case 2.2: Suppose that some matrix in has a nonzero lower right entry. That is, in we have a matrix
Since it follows that Let be an arbitrary element of . Since it follows that generates as a right ideal in .
In all cases, is a finitely generated.
|Title||example of a right noetherian ring that is not left noetherian|
|Date of creation||2013-03-22 14:16:15|
|Last modified on||2013-03-22 14:16:15|
|Last modified by||CWoo (3771)|