On rings without a certain divisibility property

Loper, Alan

doi:10.1016/0022-314x(88)90060-1

Cited by 8 publications

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“…Unit-valued polynomials and unit-valued polynomial functions have been employed in the literature to examine other mathematical objects. Loper [6] uses unit-valued polynomials for distinguishing two classes of commutative rings: D-rings and non-D-rings, where D-rings are characterized by the fact that every unit-valued polynomial is a constant. For instance, all semi-local rings (and, in particular, all finite rings) are non-D rings.…”

Section: Preliminariesmentioning

confidence: 99%

On the group of unit-valued polynomial functions

Al-Maktry

2021

AAECC

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Let R be a finite commutative ring. The set $${{\mathcal{F}}}(R)$$ F ( R ) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units $${{\mathcal{F}}}(R)^\times $$ F ( R ) × is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on $$R[x]/(x^2)=R[\alpha ]$$ R [ x ] / ( x 2 ) = R [ α ] , the ring of dual numbers over R, and show that the group $${\mathcal{P}}_{R}(R[\alpha ])$$ P R ( R [ α ] ) , consisting of those polynomial permutations of $$R[\alpha ]$$ R [ α ] represented by polynomials in R[x], is embedded in a semidirect product of $${{\mathcal{F}}}(R)^\times $$ F ( R ) × by the group $${\mathcal{P}}(R)$$ P ( R ) of polynomial permutations on R. In particular, when $$R={\mathbb{F}}_q$$ R = F q , we prove that $${\mathcal{P}}_{{\mathbb{F}}_q}({\mathbb{F}}_q[\alpha ])\cong {\mathcal{P}}({\mathbb{F}}_q) \ltimes _\theta {{\mathcal{F}}}({\mathbb{F}}_q)^\times $$ P F q ( F q [ α ] ) ≅ P ( F q ) ⋉ θ F ( F q ) × . Furthermore, we count unit-valued polynomial functions on the ring of integers modulo $${p^n}$$ p n and obtain canonical representations for these functions.

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Section: Preliminariesmentioning

confidence: 99%