9 - Fully Bounded Rings

Summary

One major obstacle to adapting commutative noetherian ring theory to the noncommutative case in general is the lack of ideals. For example, the Weyl algebras over division rings of characteristic zero are simple noetherian domains, yet their module structure is quite complicated. Thus, to derive much structure theory similar to the commutative theory, one should work in a context where a large supply of ideals is guaranteed. One such context is introduced and investigated in this chapter. The results obtained may serve to give a sample of what is known about noetherian rings satisfying a polynomial identity (P.I.), although the methods used are very different from those of P.I. theory.

• BOUNDEDNESS •

Definition. A ring R is right bounded if every essential right ideal of R contains an ideal which is essential as a right ideal.

For instance, every commutative ring is right bounded, as is every semisimple ring (since a semisimple ring has no proper essential right ideals). On the other hand, a simple ring cannot be right bounded unless it is artinian. Note that a prime ring R is right bounded if and only if every essential right ideal of R contains a nonzero ideal (recall Exercise 5A).

Definition. A ring R is right fully bounded provided every prime factor ring of R is right bounded.