order (of a ring)

The order of a ring $R$ is the order (http://planetmath.org/OrderGroup) of its additive group, i.e. (http://planetmath.org/Ie) the number of elements of $R$ . The order of $R$ can be denoted as $|R|$ . If $|R|$ is finite, then $R$ is said to be a finite ring.

This definition of order is not necessarily standard. Please see http://planetmath.org/?op=getobj&from=corrections&id=12149this 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. http://planetmath.org/?op=getobj&from=papers&id=336Cyclic Rings. Charleston, IL: Eastern Illinois University, 2004.
3.
Fine, Benjamin. “Classification of Finite Rings of Order $p^{2}$ .” 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.pdfhttp://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.htmlhttp://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.

单词	OrderofARing
释义	order (of a ring) The order of a ring $R$ is the order (http://planetmath.org/OrderGroup) of its additive group, i.e. (http://planetmath.org/Ie) the number of elements of $R$ . The order of $R$ can be denoted as $\|R\|$ . If $\|R\|$ is finite, then $R$ is said to be a finite ring. This definition of order is not necessarily standard. Please see http://planetmath.org/?op=getobj&from=corrections&id=12149this 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. http://planetmath.org/?op=getobj&from=papers&id=336Cyclic Rings. Charleston, IL: Eastern Illinois University, 2004. 3. Fine, Benjamin. “Classification of Finite Rings of Order $p^{2}$ .” 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.pdfhttp://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.htmlhttp://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.
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