B - Perfect, Semiperfect, and Semiregular Rings

Summary

The structure of a semisimple artinian ring is well known thanks to the Wedderburn–Artin theorem. Moreover, if R is right artinian then J is nilpotent and R/J is semisimple, so a major problem is to “lift” the structure of R/J to R. This leads to lifting idempotents and hence to the notion of a semiperfect ring. These rings are described in this section, together with a discussion of the important subclass of right perfect rings. Finally, the situation where R/J is regular is treated under the heading of semiregular rings. Each of these classes of rings is used extensively throughout the book.

Semiperfect Rings

Semiperfect rings are perhaps the most useful generaliz ation of the classical artinian rings. We begin by describing the simplest examples.

Proposition B.1.The following conditions are equivalent for a ring R:

1. R/J is a division ring.

2. R – J consists of units.

3. If a є R then either a or 1 – a is a unit.

4. R has a unique maximal right (respectively left) ideal.

5. J is a maximal right (respectively left) ideal.

Proof. (1)⇒(2) becausea is a unit in R whenever a + J is a unit in R/J, (2)⇒(3)⇒(4)⇒(5) are clear, and (5)⇒(1) because aR = R whenever a ∉ J by (5) because J ⊆^sm R_R.

A ring R is called local if it satisfies the conditions in Proposition B.1, and an idempotent e in R is called a local idempotent if eRe is a local ring. The following result will be used repeatedly.