Vol. 78, No. 1, 1978

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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

For a commutative ring R with identity, denote by RXthe 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 RX. 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

f = f0 +f1X + ⋅⋅⋅+ fnXn ∈ R[X ],

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 fj+1,,fn P, fjP, imply that fj is a unit modulo P.
  2. There exists a direct sum decomposition
    R = R1 ⊕ ⋅⋅⋅⊕ Rn of R

    such that if f = g1 + + gm is the decomposition of f with respect to the induced decomposition

    R [X ] = R1[X ]⊕ ⋅⋅⋅⊕ Rm [X] of R[X],

    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.

Mathematical Subject Classification 2000
Primary: 13F20
Received: 8 December 1977
Revised: 22 February 1978
Published: 1 September 1978
Robert William Gilmer, Jr.
William James Heinzer