The order of a ring is the order of its additive group, i.e. the number of elements of . The order of can be denoted as . If is finite, then is said to be a finite ring.

This definition of order is not necessarily standard. Please see this correction and the posts attached to it for more details.

This definition of order is used in the following works:

1. Angerer, Josef and Pilz, Günter. “The Structure of Near Rings of Small Order.” Computer Algebra: EUROCAM '82, European Computer Algebra Conference; Marseilles, France, April 1982. Editors: Goos, G. and Hartmanis, J. Berlin: Springer-Verlag, 1982, pp. 57-64.
2. Buck, Warren. Cyclic Rings. Charleston, IL: Eastern Illinois University, 2004.
3. Fine, Benjamin. “Classification of Finite Rings of Order .” Mathematics Magazine, vol. 66 #4. Washington, DC: Mathematical Association of America, 1993, pp. 248-252.
4. Fletcher, Colin R. “Rings of Small Order.” The Mathematical Gazette, vol. 64 #427. Leicester, England: The Mathematical Association, 1980, pp. 9-22.
5. Lam, Tsi-Yuen. A First Course in Noncommutative Rings. New York: Springer-Verlag, 2001.
6. Mitchell, James. School of Mathematics and Statistics: MT4517 Rings and Fields, Lecture Notes 1. St. Andrews, Scotland: University of St. Andrews, 2006. URL: http://www-history.mcs.st-and.ac.uk/˜jamesm/teaching/MT4517/MT4517-notes1.pdf
7. Nöbauer, Christof. Numbers of rings on groups of prime power order. Linz, Austria: Johannes Kepler Universität Linz. URL: http://www.algebra.uni-linz.ac.at/˜noebsi/ringtable.html
8. Schwabe, Eric J. and Sutherland, Ian M. “Efficient Mappings for Parity-Declustered Data Layouts.” Computing and Combinatorics: 9th Annual International Conference, COCOON 2003; Big Sky, MT, USA, July 2003; Proceedings. Editors: Warnow, Tandy and Zhu, Binhai. Berlin: Springer-Verlag, 2003, pp. 252-261.