THE LARGEST LEFT QUOTIENT RING OF A RING

摘要

The left quotient ring (i.e., the left classical ring of fractions) Q(cl)(R) of a ring R does not always exist and still, in general, there is no good understanding of the reason why this happens. In this article, existence of the largest left quotient ring Q(l)(R) of an arbitrary ring R is proved, i.e., Q(l)(R)=S-0(R)R-1 where S-0(R) is the largest left regular denominator set of R. It is proved that Q(l)(Q(l)(R))=Q(l)(R); the ring Q(l)(R) is semisimple iff Q(cl)(R) exists and is semisimple; moreover, if the ring Q(l)(R) is left Artinian, then Q(cl)(R) exists and Q(l)(R)=Q(cl)(R). The group of units Q(l)(R)* of Q(l)(R) is equal to the set {s(-1)t vertical bar s, tS(0)(R)} and S-0(R)=R Q(l)(R)*. If there exists a finitely generated flat left R-module which is not projective, then Q(l)(R) is not a semisimple ring. We extend slightly Ore's method of localization to localizable left Ore sets, give a criterion of when a left Ore set is localizable, and prove that all left and right Ore sets of an arbitrary ring are localizable (not just denominator sets as in Ore's method of localization). Applications are given for certain classes of rings (semiprime Goldie rings, Noetherian commutative rings, the algebra of polynomial integro-differential operators).