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Tensor product and theories of modules

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, England E-mail: [email protected]
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Extract

Modules over a ring R, when tensored with an (R, S)-bimodule, are converted to S-modules. Here I consider, from the standpoint of the model theory of modules, the effect of this operation. The primary motivation arises from questions concerning representation type of algebras and interpretability of modules, where such tensor functors play a key role, but this paper is devoted to more general considerations. For instance, the elementary duality of [2] and [1] is generalised here. It is also shown that, although tensor product does not preserve elementary equivalence, one can define the tensor product of two complete theories of R-modules. The results in Section 1 grew out of a number of discussions with T. Kucera.

This is our generic situation. We have an (R, S)-bimodule B and we are considering the functor –⊗RBS from Mod – R (the category of right R-modules) to Mod – S which is given on objects by MR ↦ (M ⊗RB)s and has the obvious action on morphisms. There is a somewhat more general situation: namely we may consider the effect of tensoring (T, R)-bimodules over R with B to obtain (T, S)-bimodules. For the intended applications this case is not needed. Moreover, although some results extend to this more general case, there are some which definitely do not (see the example after 2.1). Therefore we confine ourselves to the case first described. We will also consider B itself as a variable and so ask “What is the effect of tensoring R-modules with (R, S)-bimodules?”.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 2 , June 1999 , pp. 617 - 628
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[1] Herzog, I., Elementary duality, Transactions of the American Mathematical Society, vol. 340 (1993), pp. 37–69.CrossRefGoogle Scholar
[2] Prest, M., Duality and pure-semisimple rings, The Journal of the London Mathematical Society, vol. 38 (1988), pp. 403–409.Google Scholar
[3] Prest, M., Model Theory and modules, London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, 1988.Google Scholar
[4] Rothmaler, Ph., A trivial remark on purity, Proceedings of the 9th Easter conference on model theory, Gosen, Seminarber. 112 (1991), Humboldt–Universität zu Berlin, p. 127.Google Scholar
[5] Rothmaler, Ph., Mittag–Leffler modules, Annals of Pure and Applied Logic, vol. 88 (1997), pp. 227–239.Google Scholar
[6] Sabbagh, G., Aspects logiques de la pureté dans les modules, Comptes Rendus de l'Académie des Sciences, vol. 271 (1970), pp. 909–912.Google Scholar
[7] Watts, C. E., Intrinsic characterisation of some additive functors, Proceedings of the American Mathematical Society, vol. 11 (1960), pp. 5–8.CrossRefGoogle Scholar
[8] Ziegler, M., Model theory of modules, Annals of Mathematical Logic, vol. 26 (1984), no. 2, pp. 149–213.Google Scholar