The McCoy Condition on Noncommutative Rings

Abstract

McCoy proved in 1957 [12] that if a polynomial annihilates an ideal of polynomials over any ring then the ideal has a nonzero annihilator in the base ring. We first elaborate this McCoy's famous theorem further, expanding the inductive construction in the proof given by McCoy. From the proof we can naturally find nonzero c, with f(x)c = 0, in the ideal of R generated by the coefficients of g(x), when f(x), g(x) are nonzero polynomials over a commutative ring R with f(x)g(x) = 0; from which we also obtain a kind of criterion for given a polynomial to be a zero divisor. Based on these results we extend the McCoy's theorem to noncommutative rings, introducing the concept of strong right McCoyness. The strong McCoyness is shown to have a place between the reversibleness (right duoness) and the McCoyness. We introduce a simple way to construct a right McCoy ring but not strongly right McCoy, from given any (strongly) right McCoy ring. If given a ring is reversible or right duo, then the polynomial ring over it is proved to be strongly right McCoy. It is shown that the (strong) right McCoyness can go up to classical right quotient rings.