JP Journal of Algebra, Number Theory and Applications
Volume 9, Issue 2, Pages 241 - 275
(December 2007)
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COMPOSITES OF MINIMAL RING EXTENSIONS
David E. Dobbs (U. S. A.) and Jay Shapiro (U. S. A.)
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Abstract: Let R be a (commutative unital) ring. If S and T are distinct minimal ring extensions of R, their composite ST may not exist; i.e., there may not exist a (commutative unital) R-algebra U containing both S and T as R-subalgebras. We assume henceforth that such U exists. It seems natural to ask
is ST necessarily a minimal ring extension of both S and T ? If R is a field and S, T (as above) are splitting field extensions of R, the answer to
is ?yes?; without this ?splitting field? assumptionon the fields S and T, the answer is, in general, ?no?. If R is a field and either S or T is not a field, the answer to
is, in general, ?no?. Let M and N be the so-called crucial maximal ideals of R relative to S and T, respectively. If
the answer to
is ?yes?. Assume henceforth that R is a ring with von Neumann regular total quotient ring and that S and T are overrings of R. If R is integrally closed in both S and T, the answer to
is ?yes?. If
it cannot be the case that T is integral over R while R is integrally closed in S. If
with S (and hence T) integral over R, then the answer to can be ?no? and we give best-possible finite upper bounds for the cardinalities of chains of rings between S and ST and chains of rings between T and ST. |
Keywords and phrases: minimal ring extension, composite, total quotient ring, overring,von Neumann regular ring, crucial maximal ideal, integrality, flat emimorphism, idealization, algebra, Noetherian domain. |
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