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- Title
Polyfunctions over commutative rings.
- Authors
Specker, Ernst; Hungerbühler, Norbert; Wasem, Micha
- 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 ∈ 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 = ℤ / n ℤ 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 Rédei–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 Rédei–Szele theorem which are based on our results.
- Subjects
FINITE rings; FINITE fields; POLYNOMIAL rings; COMMUTATIVE rings; POLYNOMIALS
- Publication
Journal of Algebra & Its Applications, 2024, Vol 23, Issue 1, p1
- ISSN
0219-4988
- Publication type
Article
- DOI
10.1142/S0219498824500142