Ore Condition
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 domain R, and any pair a, b of non-zero elements, is the requirement that the sets aR and bR should intersect in more than the element 0. The left Ore cond...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 domain R, and any pair a, b of non-zero elements, is the requirement that the sets aR and bR should intersect in more than the element 0. The left Ore condition is defined similarly. A domain that satisfies the right Ore condition is called a right Ore domain. For every right Ore domain R, there is a unique (up to natural R-isomorphism) division ring D containing R as a subring such that every element of D is of the form rs 1, for r in R and s nonzero in R. Such a division ring D is called a ring of right fractions of R, and R is called a rightorder in D.
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