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.
Let be the ring of all matricessuch that is an integer and are rational. The claim is that is right noetherian but not left noetherian.
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 .
It is a bit harder to show that is right noetherian. The approach given here usesthe fact that a ring is right noetherian if all of its right ideals are finitely generated.
Let be a right ideal in . We show that is finitely generated bychecking all possible cases. In the first case, we assume that every matrix in hasa zero in its upper left entry. In the second case, we assume thatthere is some matrix in that has a nonzero upper left entry. The second case splitsinto 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.
Indeed, note thatis a subspace of the two dimensional vector space . So in there exist two(not necessarily linearly independent
) vectors and which span .
Now, an arbitrary elementin corresponds to the vector in and for some . Thus
and it follows that is finitely generatedby the setas a right ideal in .
CASE 2: Suppose that some matrix in has a nonzero upper left entry. Then there is a least positiveinteger occurring as the upper left entry of a matrix in . It follows that every elementof can be put into the form
By definition of , there is a matrix of the formin . Since is a right ideal in and sinceit follows thatis in . Now break off into two subcases.
case 2.1: Suppose that every matrix in has a zero in its lower right entry. Thenan 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 thatLetbe an arbitrary element of . Since it follows thatgenerates as a right ideal in .
In all cases, is a finitely generated.