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Ore condition
In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The ''right Ore condition'' for a multiplicative subset ''S'' of a ring ''R'' is that for and , the intersection . A domain that satisfies the right Ore condition is called a right Ore domain. The left case is defined similarly.
==General idea==
The goal is to construct the right ring of fractions ''R''() with respect to multiplicative subset ''S''. In other words we want to work with elements of the form ''as''−1 and have a ring structure on the set ''R''(). The problem is that there is no obvious interpretation of the product (''as''−1)(''bt''−1); indeed, we need a method to "move" ''s''−1 past ''b''. This means that we need to be able to rewrite ''s''−1''b'' as a product ''b''1''s''1−1. Suppose then multiplying on the left by ''s'' and on the right by ''s''1, we get . Hence we see the necessity, for a given ''a'' and ''s'', of the existence of ''a''1 and ''s''1 with and such that .
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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