Deductive varieties of modules and universal algebras
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 an 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.
This article is published as Hogben, Leslie, and Clifford Bergman. "Deductive varieties of modules and universal algebras." Transactions of the American Mathematical Society 289, no. 1 (1985): 303-320. DOI: 10.1090/S0002-9947-1985-0779065-X. Posted with permission.