This came up as part of a proof I'm trying to write. Suppose $P(x)$ is a polynomial of degree $n$ over a ring $R$ with identity. If its leading coefficient is a unit (i.e. has a multiplicative inverse), can the principal (two-sided) ideal $(P)$ contain nonzero polynomials of degree less than $n$? I strongly suspect the answer is no. It's clearly impossible in the case of commutative $R$, but the noncommutative case is proving surprisingly difficult because the characterization of principal ideals isn't as nice.
Can anyone spot a proof, or if I'm mistaken, a counterexample for noncommutative $R$ (preferably as elementary as possible; I'm still fairly new to ring theory)?