Geodesic
Formal power series rings over a $\pi$-domain
Journal of the European Mathematical Society, Tome 11 (2009) no. 6, pp. 1429-1443.

Voir la notice de l'article provenant de la source EMS Press

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.
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 
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     author = {Byung Gyun Kang and Dong Yeol Oh},
     title = {Formal power series rings over a $\pi$-domain},
     journal = {Journal of the European Mathematical Society},
     pages = {1429--1443},
     publisher = {mathdoc},
     volume = {11},
     number = {6},
     year = {2009},
     doi = {10.4171/jems/186},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/186/}
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Byung Gyun Kang; Dong Yeol Oh. Formal power series rings over a $\pi$-domain. Journal of the European Mathematical Society, Tome 11 (2009) no. 6, pp. 1429-1443. doi : 10.4171/jems/186. http://geodesic.mathdoc.fr/articles/10.4171/jems/186/

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