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 f_{j+1},⋯,f_{n} ∈ P,
f_{j}∉P, imply that f_{j} is a unit modulo P.
 There exists a direct sum decomposition
such that if f = g_{1} + ⋯ + g_{m} is the decomposition of f with respect to
the induced decomposition
then the leading coefficient of g_{i} is a unit of R_{i} 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.
