For a commutative ring R with
identity, denote by R⟨X⟩ the quotient ring of R[X] with respect to the regular
multiplicative system S of monic polynomials over R. The present paper
determines the group of units of the ring R⟨X⟩. This is equivalent to the
problem of determining the saturation S∗ of the multiplicative system S; by
definition, S∗ consists of all divisors of monic polynomials over R. For a nonzero
polynomial
it is shown that each of the following conditions (A) and (B) is equivalent to the
condition that f divides a monic polynomial over R (in (B), the ring R is
reduced).
The coefficients of f generate the unit ideal of R and, for each j between
0 and n and for each prime ideal P of R, the relations fj+1,⋯,fn∈ P,
fj∉P, imply that fj is a unit modulo P.
There exists a direct sum decomposition
such that if f = g1+⋯+ gm is the decomposition of f with respect to
the induced decomposition
then the leading coefficient of gi is a unit of Ri for each i.
One corollary to the preceding characterizations is that S∗ is the set of
polynomials over R with unit leading coefficient if and only if the ring R is reduced
and indecomposable.