INC-EXTENSIONS AMID ZERO-DIVISORS

Authors : David E DOBBS, Jay SHAPİRO

Pages : 102-109

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Publication Date : 2010-06-01

Article Type : Research Paper

Abstract :All rings considered are commutative with identity. Let R be a complemented ring with integral closure R0 insert ignore into journalissuearticles values(in its total quotient ring K);. Then R ⊆ S satisfies INC for each overring S of R insert ignore into journalissuearticles values(inside K); if and only if R0 is a Prüfer ring. If R0 is a Prüfer ring and T is a complemented ring that contains R as a subring such that each regular element of T is a root of a polynomial in R[X] with a regular coefficient and T is torsion-free over R, then R ⊆ T satisfies INC. As a consequence, a new generalization for rings with nontrivial zero-divisors is found of Pr¨ufer’s result on the integral closure of a Prüfer domain in a field extension of the quotient field.
Keywords : complemented ring, INC, Pr¨ufer ring, ring extension, overring, integral, algebraic, von Neumann regular ring, integral domain, torsion free, total quotient ring, flat