Rings are important structures in modern algebra. If a ring R has a multiplicative unit element 1 and every nonzero element has a multiplicative inverse, then R is called a division ring. So, all that is missing in R from being a field is the commutativity of multiplication. The best-known example of a noncommutative division ring is the ring of quaternions discovered by Hamilton. But, as the chapter title says, every such division ring must of necessity be infinite. If R is finite, then the axioms force the multiplication to be commutative.
CITATION STYLE
Aigner, M., & Ziegler, G. M. (2018). Every finite division ring is a field. In Proofs from THE BOOK (pp. 35–38). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-57265-8_6
Mendeley helps you to discover research relevant for your work.