Vol. 9, No. 5, 2015

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Factorially closed subrings of commutative rings

Sagnik Chakraborty, Rajendra Vasant Gurjar and Masayoshi Miyanishi

Vol. 9 (2015), No. 5, 1137–1158

We prove some new results about factorially closed subrings of commutative rings. We generalize this notion to quasifactorially closed subrings of commutative rings and prove some results about them from algebraic and geometric viewpoints. We show that quasifactorially closed subrings of polynomial and power series rings of dimension at most three are again polynomial (resp. power series) rings in a smaller number of variables. As an application of our results, we give a short proof of a result of Lê Dũng Tráng in connection with the Jacobian problem.

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factorially closed subring
Mathematical Subject Classification 2010
Primary: 13A05
Secondary: 13B99, 14R05
Received: 22 December 2014
Revised: 4 May 2015
Accepted: 9 May 2015
Published: 21 June 2015
Sagnik Chakraborty
School of Mathematics
Tata Institute of Fundamental Research
Homi-Bhabha Road, Navy Nagar
Mumbai 400005
Rajendra Vasant Gurjar
School of Mathematics
Tata Institute of Fundamental Research
Homi-Bhabha Road, Navy Nagar
Mumbai 400005
Masayoshi Miyanishi
Research Center for Mathematical Sciences
Kwansei Gakuin University
2-1, Gakuen
Sanda 669-1337