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Journal ArticleDOI

A division algorithm

Fred Richman
- 01 Aug 2005 - 
- Vol. 04, Iss: 04, pp 441-449
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TLDR
In this paper, it was shown that a divisibility test for polynomials over a field in an intuitionistic setting may be thought of as a kind of division algorithm.
Abstract
A divisibility test of Arend Heyting, for polynomials over a field in an intuitionistic setting, may be thought of as a kind of division algorithm. We show that such a division algorithm holds for divisibility by polynomials of content 1 over any commutative ring in which nilpotent elements are zero. In addition, for an arbitrary commutative ring R, we characterize those polynomials g such that the R-module endomorphism of R[X] given by multiplication by g has a left inverse.

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References
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Book

A Course in Constructive Algebra

TL;DR: In this paper, the authors present a generalization of the Dedekind-Hasse rings of algebraic fields to the setting of finite-rank free groups, and show that these groups can be represented by a set of polynomials.
Journal ArticleDOI

An algorithm for the divisors of monic polynomials over a commutative ring

TL;DR: Gilmer and Heinzer as mentioned in this paper proved that given a reduced ring R, a polynomial f can be divisible if and only if there exists a direct sum decomposition of R = R0 ⊕ Rm (m ≤ deg f), associated to a fundamental system of idempotents e0, …, em, such that the component of f in each Ri[X] has degree coefficient which is a unit of Ri.