Polyfunctions over commutative rings
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2024-01
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Journal Article
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Abstract
A function f : R -> R, where R is a commutative ring with unit element, is called polyfunction if it admits a polynomial representative p is an element of R[x]. Based on this notion, we introduce ring invariants which associate to R the numbers s(R) and s(R '; R), where R ' is the subring generated by 1. For the ring R = Zeta/n Zeta the invariant s(R) coincides with the number theoretic Smarandache or Kempner function s(n). If every function in a ring R is a polyfunction, then R is a finite field according to the Redei-Szele theorem, and it holds that s(R) = |R|. However, the condition s(R) = |R| does not imply that every function f : R -> R is a polyfunction. We classify all finite commutative rings R with unit element which satisfy s(R) = |R|. For infinite rings R, we obtain a bound on the cardinality of the subring R ' and for s(R '; R) in terms of s(R). In particular we show that |R '|<= s(R)!. We also give two new proofs for the Redei-Szele theorem which are based on our results.
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published
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23 (1)
Pages / Article No.
2450014
Publisher
World Scientific
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Software
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Subject
Polyfunctions; Kempner function
Organisational unit
03874 - Hungerbühler, Norbert / Hungerbühler, Norbert