In this paper we study basic division properties in the ring of regular quaternionic polynomials.We obtain a Bezout-like theoremand we calculate the module syzygy for any vector of polynomials. © 2009 Elsevier Ltd. All rights reserved.

the aim of this paper is to study orders over a valuation ring v with arbitrary rank in acentral simple f-algebra q. the relation between all of the orders is explained with a diagram. it is thenshown that inside bezout order, extremal v-orders are precisely semi-hereditary. in the last section, theeffect of henselization on maximal and semi-hereditary orders is examined.

An element of a ring $R$ is called clear if it sum unit-regular and unit. associative each its elements clear.In this paper we defined rings extended many results to wider class. Finally, proved that commutative Bezout domain an elementary divisor only every full $2\times 2$ matrix over nontrivially clear.

We present a formalization of the factor theorem for univariate polynomials, also called the (little) Bezout theorem: Let r belong to a commutative ring L and p(x) be a polynomial over L. Then x− r divides p(x) iff p(r) = 0. We also prove some consequences of this theorem like that any non zero polynomial of degree n over an algebraically closed integral domain has n (non necessarily distinct) ...

We present a formalization of the factor theorem for univariate polynomials, also called the (little) Bezout theorem: Let r belong to a commutative ring L and p(x) be a polynomial over L. Then x− r divides p(x) iff p(r) = 0. We also prove some consequences of this theorem like that any non zero polynomial of degree n over an algebraically closed integral domain has n (non necessarily distinct) ...

The Bezout-Inequality, an afine version (not in&ding multiplicities) of the classical Bezout-Theorem is derived for applications in algebraic complexity theory. Upper hounds for the cardinality and number of sets definable by first order formulas over algebraically closed fields are given. This is used for fast quantifier elimination in algebraically closed fields.