Abstract: Let be a polynomial in several indeterminates with coefficients in an integral domain R with quotient field K. We prove that the principal ideal generated by f in the polynomial ring is prime iff f is irreducible over K and where A is the content of f. We also prove that if is such that and is a primitive polynomial in the sense that only a unit of R can divide each coefficient of g, then fg will be primitive.
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13F20
Retrieve articles in all journals with MSC: 13F20
Keywords: Gauss' Lemma, integral domain, content of polynomial, principal prime ideal, primitive polynomial, greatest common divisor, unique factorization domain, integrally closed domain
Article copyright: © Copyright 1972 American Mathematical Society