Dominant modules and finite localizations
Edgar Andrews Rutter · Tohoku Mathematical Journal · 1975
Colby and Rutter [6] proved that a ring R contains an essential left socle and an essential right socle and has a two sided semi-simple maximal (complete) quotient ring if and only if R is a QF-Z ring with zero right singular ideal.Utilizing the fact that a left perfect ring contains a minimal dense right ideal, Storrer [29] proved that a left and right artinian ring R has a two sided quasi-Frobenius maximal quotient ring if and only if R is a QF-Z ring whose minimal dense right ideal is projective.This paper was motivated by the desire to obtain a common generalization of these results.Kato's [13] notion of a dominant module proved useful in this connection and the first section of this paper is devoted to results concerning dominant modules.With regard to the problem posed above, the most relevant are:A finitely generated projective left module is a dominant left module if and only if its trace ideal is a minimal dense right ideal.The minimal faithful left module over a left QF-3 ring is a dominant left module.This section also contains results not directly related to our principal objective.Also useful in this connection is Silver's [27] concept of a finite right localization.In the second section, we characterize those rings R whose maximal ring of right quotients is a finite right localization of R and belongs to one of the following classes of rings: right S-rings, semi-simple rings, right self injective right cogenerator rings, and right self injective rings.If the maximal ring of right quotients of a left perfect ring belongs to any of the first three of these classes, it is necessarily a finite right localization of R. Thus in most cases the results of this section generalize results of Storrer [29].In some instances they sharpen Storrer's results even in case R is left perfect.The above results are used to prove that the following statements are equivalent: R has a two sided maximal quotient ring which is a cogenerator ring and projective both as a left and a right JS-module.R contains a minimal dense left and a minimal dense right ideal and