Rings with the minimum condition for principal right ideals have the maximum condition for principal left ideals

Abstract

Bass [1] called a ring perfect if each left R-module has a projective cover. He showed that a ring was perfect if and only if it satisfied the descending chain condition for principal right ideals; furthermore, he showed that this was equivalent to the Jacobson radical N being T-nilpotent and R/N having the descending chain condition. An ideal is called T-nilpotent if for each sequence {r,} of elements of the ideal there is an integer k such that the product r 1 r 2 ... r k is zero. Thus the rings with descending chain condition are perfect rings; in this case the radical is nilpotent. There are examples of perfect rings for which the radical is not nilpotent. The author is indebted to his colleague Richard Courter for many encouraging conversations; in particular, for his matrix proof that full matrix rings over perfect local rings have the maximum condition for left principal ideals. The corresponding property for perfect rings was a natural conjecture. An examination of the "matrix" rings