On 𝑓-rings with the ascending chain condition

Abstract

Introduction. In [1] Birkhoff and Pierce obtain the structure of f-rings2 which have no nonzero nilpotent elements and satisfy the descending chain condition for i-ideals. More recently, D. G. Johnson [4] gives the structure of J-semi-simple f-rings (?2) with the descending chain condition for i-ideals. In this note our principal aim is to give the structure of f-rings with various ascending chain conditions. We first show (Theorem 1) that in f-rings the ascending and descending chain conditions for closed i-ideals are equivalent and that an f-ring with these conditions can be characterized as a subdirect sum of finitely many totally ordered rings. Next (Theorem 2) we specialize to the case of f-rings with no nonzero nilpotent elements. In ?2 we consider J-semi-simple f-rings. For these f-rings we show (Theorem 4) that the ascending and descending chain conditions for i-ideals and for closed i-ideals are all equivalent. In [3] Goldie proves that a semi-simple ring with the ascending chain condition for ideals is a subdirect sum of a finite number of semi-simple prime rings. An examination of the proof of this result shows that he proves even more, namely, that a semi-prime ring with the ascending chain condition for annihilator ideals is a subdirect sum of a finite number of prime rings. The results of this note provide f-ring analogues of the results of [3], and the techniques we employ are patterned after those of Goldie.