The Gaussian Property for Rings and Polynomials

Abstract. The content of a polynomial f over a commutative ring R is the ideal c(f) of R generated by the coefficients of f. If c(fg) = c(f)c(g) for each polynomial g ∈ R[x], then f is said to be Gaussian. The ring R is Gaussian if each polynomial in R[x] is Gaussian. It is known that f is Gaussian if c(f) is locally principal. The converse is established for polynomials over reduced rings. Also, if the square of the nilradical is zero, then R is Gaussian if and only if the square of each finitely generated ideal is locally principal. 1