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Preservation of elementary equivalence under scalar extension1

Published online by Cambridge University Press:  12 March 2014

Bruce I. Rose
Bruce I. Rose*
Affiliation:
University of Notre Dame, Notre Dame, Indiana 46556
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Extract

In this note we show that taking a scalar extension of two elementarily equivalent finite-dimensional algebras over the same field preserves elementary equivalence. The general question of whether or not tensor product preserves elementary equivalence was originally raised in [4]. In [3] Feferman relates an example of Ersov which answers the question negatively. Eklof and Olin [7] also provide a counterexample to the general question in the context of two-sorted structures. Thus the result proved below is a partial positive answer to a general question whose status has been resolved negatively. From the viewpoint of applied model theory it seems desirable to find contexts in which positive statements of preservation can be obtained. Our result does have an application; a corollary to it increases our understanding of what it means for two division algebras to be elementarily equivalent.

All algebras are finite-dimensional algebras over fields. All algebras contain an identity element, but are not necessarily associative.

Recall that the center of a not necessarily associative algebra A is the set of elements which commute and “associate” with all elements of A. The notion of a scalar extension is an important one in algebra. If A is an algebra over F and G is an extension field of F, then the scalar extension of A by G is the algebra A ⊗F G.

Keywords

Elementary equivalencescalar extensionfinite-dimensional algebranonassociative algebradivision algebra
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 4 , December 1982 , pp. 734 - 738
Copyright
Copyright © Association for Symbolic Logic 1982

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Footnotes

1

These results were presented at the AMS meeting in Providence, R.I., August, 1978, at the special session on problems in logic arising from mathematics.

References

REFERENCES

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