Let R and S be unital algebras. We show that if X is a d-free subset of R and S is finite dimensional, then the set X={x⊗s|x∈X,s∈S} is a d-free subset of the algebra R⊗S. The assumption that S is ...finite dimensional turns out to be necessary in general. However, we show that some important functional identities have only standard solutions on X even when S is infinite dimensional.
The fundamental theorem on functional identities states that a prime ring
R
with
deg
(
R
)
≥
d
is a
d
-free subset of its maximal left ring of quotients
Q
m
l
(
R
). We consider the question whether ...the same conclusion holds for symmetric rings of quotients. This indeed turns out to be the case for the maximal symmetric ring of quotients
Q
m
s
(
R
), but not for the symmetric Martindale ring of quotients
Q
s
(
R
). We show, however, that if the maps from the basic functional identities have their ranges in
R
, then the maps from their standard solutions have their ranges in
Q
s
(
R
). We actually prove a more general theorem which implies both aforementioned results. Its proof is somewhat shorter and more compact than the standard proof used for establishing
d
-freeness in various situations.
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