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