Premium
Projective modules over polynomial rings: a constructive approach
Author(s) -
Barhoumi S.,
Lombardi H.,
Yengui I.
Publication year - 2009
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200610772
Subject(s) - krull dimension , mathematics , constructive , dimension (graph theory) , constructive proof , dimension theory (algebra) , finitely generated abelian group , pure mathematics , discrete mathematics , polynomial ring , domain (mathematical analysis) , ring (chemistry) , polynomial , algebra over a field , mathematical analysis , noetherian , computer science , chemistry , organic chemistry , process (computing) , operating system
We give a constructive proof of the fact that finitely generated projective modules over a polynomial ring with coefficients in a Prüfer domain R with Krull dimension ≤ 1 are extended from R . In particular, we obtain constructively that finitely generated projective R [ X 1 , …, X n ]‐modules, where R is a Bezout domain with Krull dimension ≤ 1, are free. Our proof is essentially based on a dynamical method for decreasing the Krull dimension and a constructive rereading of the original proof given by Maroscia and Brewer & Costa. Moreover, we obtain a simple constructive proof of a result due to Lequain and Simis stating that finitely generated modules over R [ X 1 , …, X n ], n ≥ 2, are extended from R if and only if this holds for n = 1, where R is an arithmetical ring with finite Krull dimension (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)