Chapter 2 - Noncommutative Semiperfect and Semiprime

Summary

This chapter contains results on the right FPF rings which are semiperfect. The semiperfect rings have two properties which facilitate the inquiry: I) They have a basic ring and primitive idempotents, II) The Krull-Schmidt theorem holds for direct sums of principal indecomposables. The first theorem states that any semiperfect right FP²F (or FP2 F) ring R is a direct sum of uniform right prindecs (= principal indecomposable right ideals), the basic ring RO is strongly right bounded (Compare Theorem 1.7B: any right PF ring has basic ring R₀ which is strongly bounded on both sides) and is a direct summand of every finitely generated (presented) faithful right module. Moreover, if R or R/rad R is prime, then R is a full matrix ring R₀ over a n local ring prime right (Corollaries A which is right FPF (FP 2F) and if R is FPF, then A is a two-sided valuation domain 2.lC and 2.11E).

A semiperfect right self-injective ring R with strongly right bounded basic ring is right FPF (Theorem 2.2), moreover, the converse holds provided that for each right prindec eR, every element of e(radR)e is a zero divisor of eRe, e.g. if rad R is nil (Corollary 2.2B). Although in general the basic ring of a semiperfect FPF ring is not a product of local rings, that of any semiperfect right CFPF ring R is. (See Theorem 2.5, and 2.7 for a converse.)