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The representation theories of elementarily equivalent rings

Published online by Cambridge University Press:  12 March 2014

Mike Prest
Mike Prest*
Affiliation:
Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK, E-mail: mprest@ma.man.ac.uk
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Extract

Two rings are elementarily equivalent if and only if they satisfy the same sentences in the language of rings. To some extent this is reflected in the representation theories of the two rings (one may see both positive and negative results along these lines in [5]). Here we concentrate on the case of a ring S and an elementary subring R of S.

We show that if R is an elementary subring of S then the lattice of pp formulas for R-modules embeds in that for S-modules. These lattices, moreover, have the same finite sub-posets. From this we derive some corollaries. Then we consider to what extent the compositions of induction and restriction between R- and S-modules preserve elementary equivalence.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 2 , June 1998 , pp. 439 - 450
Copyright
Copyright © Association for Symbolic Logic 1998

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References

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