Premium
An algorithm for the divisors of monic polynomials over a commutative ring
Author(s) -
Yengui Ihsen
Publication year - 2003
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310108
Subject(s) - monic polynomial , mathematics , ring (chemistry) , laurent polynomial , polynomial , unit (ring theory) , commutative ring , commutative property , degree (music) , component (thermodynamics) , pure mathematics , discrete mathematics , combinatorics , algebra over a field , mathematical analysis , acoustics , thermodynamics , chemistry , organic chemistry , physics , mathematics education
Gilmer and Heinzer proved that given a reduced ring R , a polynomial f divides a monic polynomial in R [ X ] if and only if there exists a direct sum decomposition of R = R 0 ⊕ … ⊕ R m ( m ≤ deg f ), associated to a fundamental system of idempotents e 0 , … , e m , such that the component of f in each R i [ X ] has degree coefficient which is a unit of R i . We propose to give an algorithm to explicitly find such a decomposition. Moreover, we extend this result to divisors of doubly monic Laurent polynomials.