Abstract
We show that the quasiequational theory of a relatively congruence modular quasivariety of left R-modules is determined by a two-sided ideal in R together with a filter of left ideals. The two-sided ideal encodes the identities that hold in the quasivariety, while the filter of left ideals encodes the quasiidentities. The filter of left ideals defines a generalized notion of torsion.
It follows from our result that if R is left Artinian, then any relatively congruence modular quasivariety of left R-modules is axiomatizable by a set of identities together with at most one proper quasiidentity, and if R is a commutative Artinian ring then any relatively congruence modular quasivariety of left R-modules is a variety.
Dedicated to Don Pigozzi
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Kearnes, K.A. (2018). Relatively congruence modular quasivarieties of modules. In: Czelakowski, J. (eds) Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science. Outstanding Contributions to Logic, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-319-74772-9_8
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DOI: https://doi.org/10.1007/978-3-319-74772-9_8
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