Formal power series rings over a $\pi$-domain

Byung Gyun Kang ; Dong Yeol Oh

Journal of the European Mathematical Society, Tome 11 (2009) no. 6, pp. 1429-1443

Résumé

Let R be an integral domain, X be a set of indeterminates over R, and R[[X]]3​ be the full ring of formal power series in X] over R. We show that the Picard group of R[[X]]3​ is isomorphic to the Picard group of R. An integral domain is called a π-domain if every principal ideal is a product of prime ideals. An integral domain is a π-domain if and only if it is a Krull domain that is locally a unique factorization domain. We show that R[[X]]3​ is a π-domain if R[[X1​,...,Xn​]] is a π-domain for every n≥1. In particular, R[[X]]3​ is a π-domain if R is a Noetherian regular domain. We extend these results to rings with zero-divisors. A commutative ring R with identity is called a π-ring if every principal ideal is a product of prime ideals. We show that R[[X]]3​ is a π-ring if R is a Noetherian regular ring.

Détail

DOI : 10.4171/jems/186

Classification : 13-XX, 00-XX
Keywords: Krull domain, π-domain, unique factorization domain, formal power series ring, invertible ideal, class group, Picard group