Subrings invariant under endomorphisms

Shulim Kaliman

doi:10.1016/S0021-8693(02)00680-4

Subrings invariant under endomorphisms

Shulim Kaliman

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let S and R be the rings of regular functions on affine algebraic varieties over a field of characteristic 0, R be embedded as a subring in S, and F: S → S be an endomorphism such that F(R) ⊂ R. Suppose that every ideal of height 1 in R generates a proper ideal in S, and the spectrum of R has no self-intersection points. We show that if F is an automorphism so is F _R : R → R. When R and S have the same transcendence degree then the fact that F _R is an automorphisms implies that F is an automorphism.

Original language	English (US)
Pages (from-to)	31-43
Number of pages	13
Journal	Journal of Algebra
Volume	261
Issue number	1
DOIs	https://doi.org/10.1016/S0021-8693(02)00680-4
State	Published - Mar 1 2003

ASJC Scopus subject areas

Algebra and Number Theory

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title = "Subrings invariant under endomorphisms",

abstract = "Let S and R be the rings of regular functions on affine algebraic varieties over a field of characteristic 0, R be embedded as a subring in S, and F: S → S be an endomorphism such that F(R) ⊂ R. Suppose that every ideal of height 1 in R generates a proper ideal in S, and the spectrum of R has no self-intersection points. We show that if F is an automorphism so is F R : R → R. When R and S have the same transcendence degree then the fact that F R is an automorphisms implies that F is an automorphism.",

author = "Shulim Kaliman",

note = "Funding Information: E-mail address: kaliman@math.miami.edu. 1 The author was partially supported by the NSA grant MDA904-00-1-0016.",

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N2 - Let S and R be the rings of regular functions on affine algebraic varieties over a field of characteristic 0, R be embedded as a subring in S, and F: S → S be an endomorphism such that F(R) ⊂ R. Suppose that every ideal of height 1 in R generates a proper ideal in S, and the spectrum of R has no self-intersection points. We show that if F is an automorphism so is F R : R → R. When R and S have the same transcendence degree then the fact that F R is an automorphisms implies that F is an automorphism.

AB - Let S and R be the rings of regular functions on affine algebraic varieties over a field of characteristic 0, R be embedded as a subring in S, and F: S → S be an endomorphism such that F(R) ⊂ R. Suppose that every ideal of height 1 in R generates a proper ideal in S, and the spectrum of R has no self-intersection points. We show that if F is an automorphism so is F R : R → R. When R and S have the same transcendence degree then the fact that F R is an automorphisms implies that F is an automorphism.

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