Division in Abstract Algebra
In abstract algebras such as matrix algebras and quaternion algebras, fractions such as are typically defined as or where is presumed an invertible element (i.e., there exists a multiplicative inverse such that where is the multiplicative identity). In an integral domain where such elements may not exist, division can still be performed on equations of the form or by left or right cancellation, respectively. More generally "division" in the sense of "cancellation" can be done in any ring with the aforementioned cancellation properties. If such a ring is finite, then by an application of the pigeonhole principle, every nonzero element of the ring is invertible, so division by any nonzero element is possible in such a ring. To learn about when algebras (in the technical sense) have a division operation, refer to the page on division algebras. In particular Bott periodicity can be used to show that any real normed division algebra must be isomorphic to either the real numbers R, the complex numbers C, the quaternions H, or the octonions O.
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