nested ideals in von Neumann regular ring
Let be an ideal of the von Neumann regular ring . Then itself is a von Neumann regular ring and any ideal of is likewise an ideal of .
If , then for some . Setting we see that belongs to the ideal and
Secondly, we have to show that whenever and , then both and lie in . Now, because is an ideal of . Thus there is an element in satisfying . Since belongs to and is assumed to be an ideal of , we conclude that the product must lie in , i.e. . Similarly it can be shown that . ∎
- 1 David M. Burton: A first course in rings and ideals. Addison-Wesley. Reading, Massachusetts (1970).