Is the class of polynomial rings an axiomatizable class?

Question

I define a polynomial ring to be a ring $(R;+,-,*,0,1)$ which is isomorphic to the ring of polynomials in one variable of a non-trivial commutative ring with unity. Is the class of polynomial rings a first-order axiomatizable class? If so, what are the necessary axioms to axiomatize it? Also, bonus question, if that class is not axiomatizable, is the first-order theory associated with that class a finitely axiomatizable theory?