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

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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.

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