TITLE: Minimal generating sets of free algebras of varieties
QUESTION [1 upvotes]: Let $V$ be a variety and $F$ be a relatively free algebra in $V$. Suppose $X$ is a minimal generating set for $F$. Under what conditions we can deduce that $X$ is a free basis of $F$?

REPLY [1 votes]: This is a partial answer.
The question has at least two possible meanings. Q1 = the question above with "minimal generating set" interpreted to mean "generating set of least cardinality". Q2 = the question above with "minimal generating set" interpreted to mean "generating set that is minimal under inclusion".
The answer to both questions for varieties of $R$-modules is easy to work out.
A1: The variety of all $R$-modules has the property that any least cardinality generating set $X'$ in a finitely generated free module $F_R(X)$ is a free generating set iff $M_n(R)$ is Dedekind finite for each $n$. Such rings are called "stably finite".
A2: The variety of all $R$-modules has the property that any $\subseteq$-minimal generating set $X'$ in a finitely generated free module $F_R(X)$ is a free generating set iff $R$ is a local ring (the nonunits form a 2-sided ideal).
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