Deductive varieties of modules and universal algebras

Hogben, Leslie; Bergman, Clifford

doi:10.1090/S0002-9947-1985-0779065-X

Deductive varieties of modules and universal algebras

Authors: Leslie Hogben and Clifford Bergman
Journal: Trans. Amer. Math. Soc. 289 (1985), 303-320
MSC: Primary 08C15; Secondary 16A35
DOI: https://doi.org/10.1090/S0002-9947-1985-0779065-X
MathSciNet review: 779065
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Abstract: A variety of universal algebras is called deductive if every subquasivariety is a variety. The following results are obtained: (1) The variety of modules of an Artinian ring is deductive if and only if the ring is the direct sum of matrix rings over local rings, in which the maximal ideal is principal as a left and right ideal. (2) A directly representable variety of finite type is deductive if and only if either (i) it is equationally complete, or (ii) every algebra has an idempotent element, and a ring constructed from the variety is of the form (1) above.

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