On the divisors of monic polynomials over a commutative ring

Robert William Gilmer, Jr. and William James Heinzer

Vol. 78 (1978), No. 1, 121–131

DOI: 10.2140/pjm.1978.78.121

Abstract

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).

1. 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.
2. There exists a direct sum decomposition

   such that if f = g₁ + ⋯ + 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.

Mathematical Subject Classification 2000

Primary: 13F20

Milestones

Received: 8 December 1977

Revised: 22 February 1978

Published: 1 September 1978