Residual bounds for varieties of modules

Abstract

In [3] and also in [4], Freese and McKenzie prove that a finitely generated congruence modular variety is residually small if and only if it is residually less than n for some n < ~o. In fact, they showed that ifA generates a residually small congruence modular variety and [A] = m, then "//(A) is residually less than the bound l! + m where l = m .... 3. They obtained this result by associating to ~'(A) a finite ring with unit, R = R(~(A), m), and relating the size of the subdirectly irreducible algebras in "f/"(A) to the size of subdirectly irreducible R-modules. Their final bound is based on two estimates. First they estimate that tR[ < / = m m'~ ~. Then they show that ifR is a ring of cardinality l, the subdirectly irreducible R-modules are bounded by l!. Using these estimates and applying modular commutator theory leads them to their result. In [1], we asked whether there is a function f such that for any residually small variety f~ with the CEP that is generated by an algebra of cardinality m, one has that ~ is residually less than f(m). This question was answered affimatively by E. Kiss in [2]. His proof does not produce a suitable function and no choice for f is known. We will show that, if R is finite, no subdirectly irreducible R-module can have larger cardinality than IR]. This has the effect of removing the factorial sign in Freese and McKenzie's result. It has a second consequence of providing a tight bound on the size of subdirectly irreducibles in finitely generated, congruence modular varieties with the CEP.