2011/01/26 by V. V. Bavula, Bavula, V. V.
Mathematics · #13N10 #16P40 #16S32 #16U20 #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.RA #msc:13N10 #msc:16P40 #msc:16S32 #msc:16U20
paper · pdf · doi:10.48550/arxiv.1101.5107
32 pages
arxiv created 2011/01/26 · openalex publication_date 2011/01/26 · arxiv updated 2011/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The left quotient ring (i.e. the left classical ring of fractions) Qcl(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 paper, it is proved existence of the largest left quotient ring Ql(R), i.e. Ql(R) = S0(R)-1R where S0(R) is the largest left regular denominator set of R. It is proved that Ql(Ql(R))=Ql(R); the ring Ql(R) is semi-simple iff Qcl(R) exists and is semi-simple; moreover, if the ring Ql(R) is left artinian then Qcl(R) exists and Ql(R) = Qcl(R). The group of units Ql(R)^* of Ql(R) is equal to the set \s-1 t | s,t∈ S0(R)\ and S0(R) = R∩ Ql(R)^*. If there exists a finitely generated flat left R-module which is not projective then Ql(R) is not a semi-simple 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 (semi-prime Goldie rings, Noetherian commutative rings, the algebra of polynomial integro-differential operators).