Ore domain

Let $R$ be a domain (http://planetmath.org/IntegralDomain).We say that $R$ is a right Ore domainif any two nonzero elements of $R$ have a nonzero common right multiple,i.e. for every pair of nonzero $x$ and $y$ ,there exists a pair of elements $r$ and $s$ of $R$ such that $xr=ys\eq 0$ .

This condition turns out to be equivalentto the following conditions on $R$ when viewed as a right $R$ -module:
(a) $R_{R}$ is a uniform module.
(b) $R_{R}$ is a module of finite rank.

The definition of a left Ore domain is similar.

If $R$ is a commutative domain (http://planetmath.org/IntegralDomain),then it is a right (and left) Ore domain.

单词	OreDomain
释义	Ore domain Let $R$ be a domain (http://planetmath.org/IntegralDomain).We say that $R$ is a right Ore domainif any two nonzero elements of $R$ have a nonzero common right multiple,i.e. for every pair of nonzero $x$ and $y$ ,there exists a pair of elements $r$ and $s$ of $R$ such that $xr=ys\eq 0$ . This condition turns out to be equivalentto the following conditions on $R$ when viewed as a right $R$ -module: (a) $R_{R}$ is a uniform module. (b) $R_{R}$ is a module of finite rank. The definition of a left Ore domain is similar. If $R$ is a commutative domain (http://planetmath.org/IntegralDomain),then it is a right (and left) Ore domain.
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