Abstract
Let F be a commutative ring, F(X, Y) the ring of polynomials over F in two noncommuting indeterminates, and F[X, Y] the ring of polynomials in two commuting indeterminates. Consider the family of polynomials in F(X, Y) which have trivial image under the natural F-algebra homomorphism from F(X, Y) to F[X, Y]. Such a polynomial will be said to have type a if each of its monomials has length at least 3. If, moreover, each monomial has X-degree at least 2, then the polynomial is by definition of type /3. Let R be an F-algebra. Following Streb [5], we call R an (F,a)-ring ((F, /3 )-ring) if, for each x, y E R, there exists an f (X, Y) of type a (type /3) for which
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