Algebra pp 388-401 | Cite as

Noetherian Semiprime Rings

  • Carl Faith
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 190)


A ring S is a (classical) right quotient ring of a subring T if every regular element aT has an inverse in S and
$$ S = \{ a{b^{ - 1}}|a,b \in T,b\;{\text{reular}}\} $$
Then T is an order in S (cf. 7.21). The following condition is necessary and sufficient for a ring T to possess a classical quotient ring: If a, bT, and if b is regular, then there exist a 1, b 1T, b 1 regular, such that ab 1 = ba 1 (see 9.1). If T is commutative, this condition is automatic, and if T is a domain, this is the Ore condition.


Prime Ring Regular Element Noetherian Ring Matrix Ring Quotient Ring 
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Copyright information

© Springer-Verlag, Berlin · Heidelberg 1973

Authors and Affiliations

  • Carl Faith
    • 1
  1. 1.Rutgers UniversityNew BrunswickUSA

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