Bavula, V.V. (2016) The Largest Left Quotient Ring of a Ring. Communications in Algebra, 44 (8). pp. 3219-3261. ISSN 0092-7872
Abstract
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 article, existence of the largest left quotient ring Ql(R) of an arbitrary ring R is proved, 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 semisimple iff Qcl(R) exists and is semisimple; 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−1t | 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 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).
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © Taylor & Francis Group, LLC. This is an author produced version of a paper subsequently published in Communications in Algebra. Uploaded in accordance with the publisher's self-archiving policy. |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 30 Nov 2017 09:33 |
Last Modified: | 30 Nov 2017 12:09 |
Published Version: | https://doi.org/10.1080/00927872.2015.1082577 |
Status: | Published |
Publisher: | Taylor & Francis |
Refereed: | Yes |
Identification Number: | 10.1080/00927872.2015.1082577 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:124646 |