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1973•Transactions of the American Mathematical SocietyRequires access

Primary ideals in rings of analytic functions

R. Douglas Williams

Open publisher page 4 citations

Abstract

Let A be the ring of all analytic functions on a connected, noncompact Riemann surface. We use the valuation theory of the ring A as developed by N. L. Alling to analyze the structure of the primary ideals of A. We characterize the upper and lower primary ideals of A and prove that every nonprime primary ideal of A is either an upper or a lower primary ideal. In addition we give some necessary and sufficient conditions for certain ideals of A to be intersections of primary ideals.

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What this paper is about

Let A be the ring of all analytic functions on a connected, noncompact Riemann surface. We use the valuation theory of the ring A as developed by N. L. Alling to analyze the structure of the primary ideals of A. We characterize the upper and lower primary ideals of A and prove that every nonprime primary ideal of A is either an upper or a lower primary ideal. In addition we give some necessary and sufficient conditions for certain ideals of A to be intersections of primary ideals.

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Available abstract

Let A be the ring of all analytic functions on a connected, noncompact Riemann surface. We use the valuation theory of the ring A as developed by N. L. Alling to analyze the structure of the primary ideals of A. We characterize the upper and lower primary ideals of A and prove that every nonprime primary ideal of A is either an upper or a lower primary ideal. In addition we give some necessary and sufficient conditions for certain ideals of A to be intersections of primary ideals.

Key concepts: Mathematics, Discrete valuation ring, Ideal (ethics), Combinatorics, Pure mathematics, Epistemology, Philosophy

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