  We consider applications of a finitary version of the Affine Representability
theorem, which follows from recent work of Belov-Kanel, Rowen, and Vishne.
Using this result we are able to show that when given a finite set of
polynomial identities, there is an algorithm that terminates after a finite
number of steps which decides whether these identities force a ring to be
commutative. We then revisit old commutativity theorems of Jacobson and
Herstein in light of this algorithm and obtain general results in this vein. In
addition, we completely characterize the homogeneous multilinear identities
that imply the commutativity of a ring.
