Pseudobases in direct powers of an algebra

Abstract:

A subset $P$ of an abstract algebra $A$ is a pseudobasis if every function from $P$ into $A$ extends uniquely to an endomorphism on $A$. $A$ is called $\kappa$-free has a pseudobasis of cardinality $\kappa$; $A$ is minimally free if $A$ has a pseudobasis. (The $0$-free algebras are "rigid" in the strong sense; the $1$-free groups are always abelian, and are precisely the additive groups of $E$-rings.) Our interest here is in the existence of pseudobases in direct powers ${A^I}$ of an algebra $A$. On the positive side, if $A$ is a rigid division ring, $\kappa$ is a cardinal, and there is no measurable cardinal $\mu$ with $|A| < \mu \leq \kappa$, then ${A^I}$ is $\kappa$-free whenever $|I| = |{A^\kappa }|$. On the negative side, if $A$ is a rigid division ring and there is a measurable cardinal $\mu$ with $|A| < \mu \leq |I|$, then ${A^I}$ is not minimally free.