  It is shown how to compute quotients efficiently in non-commutative
univariate polynomial rings. This extends earlier work where efficient generic
quotients were studied with a primary focus on commutative domains. Fast
algorithms are given for left and right quotients of polynomials where the
variable commutes with coefficients. These algorithms are based on the concept
of the ``whole shifted inverse'', which is a specialized quotient where the
dividend is a power of the polynomial variable. It is also shown that when the
variable does not commute with coefficients, that is for skew polynomials, left
and right whole shifted inverses are defined and may be used to compute right
and left quotients. In this case their computation is not asymptotically fast,
but once obtained, they may be used to compute multiple quotients, each with
one multiplication. Examples are shown of polynomials with matrix coefficients,
differential operators and difference operators. In addition, a
proof-of-concept generic Maple implementations is given.

Symbolic Computation