JournalsjemsVol. 11, No. 6pp. 1429–1443

Formal power series rings over a <em>π</em>-domain

  • Byung Gyun Kang

    Pohang University of Science and Technology, South Korea
  • Dong Yeol Oh

    National Institute for Mathematical Sciences, Daejeon, South Korea
Formal power series rings over a <em>π</em>-domain cover
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Abstract

Let R be an integral domain, Χ be a set of indeterminates over R, and R[[Χ]]3 be the full ring of formal power series in Χ over R. We show that the Picard group of R[[Χ]]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[[Χ]]3 is a π-domain if R[[_Χ_1 , . . . , Χ__n]] is a π-domain for every n ≥ 1. In particular, R[[Χ]]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[[Χ]]3 is a π-ring if R is a Noetherian regular ring.

Cite this article

Byung Gyun Kang, Dong Yeol Oh, Formal power series rings over a <em>π</em>-domain. J. Eur. Math. Soc. 11 (2009), no. 6, pp. 1429–1443

DOI 10.4171/JEMS/186