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# order (of a ring)

The *order* of a ring $R$ is the order of its additive group, i.e. 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 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 $p^{2}$.”

*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.pdf7. 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.html8. 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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## Comments

## "order of a ring" with a vengeance

Due to the correction http://planetmath.org/?op=getobj&from=corrections&id=12149 and the posts that it generated, I have decided to add this definition. I also wanted to add this because I think that, to a person who is just learning abstract algebra, it might be confusing to click on "order" in a phrase like "order of a ring" and be redirected to an entry on order of a group. I made sure to include a disclaimer that the definition is not standard, and I have also cited sources in which this definition is used. (I was able to verify what CWoo said about the source that he provided. Thanks Chi!)

There are two things that I would *definitely* like to see edited:

1. If you know of any other sources that use "order of a ring", please add them! I do not intend to have an all inclusive list of such sources, but having some sources provided shows that the term is used, even if it is not standard.

2. I definitely do not want this entry to cause erroneous links! Please edit the linking policy as you see fit.

Warren

## Re: "order of a ring" with a vengeance

http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_algebraist_2006;task=show_m...

http://books.google.com/books?id=iDHf4-jma1oC&pg=PA257&lpg=PA257&dq=%22o...

p.257, Computing and Combinatorics: 9th Annual International Conference, COCOON 2003, Big Sky, MT, USA

www-history.mcs.st-and.ac.uk/~jamesm/teaching/MT4517/MT4517-notes1.pdf

Remark 3.4.6

School of Mathematics and Statistics

MT4517 Rings & Fields

Lecture Notes 1

http://www.michmaa.org/meeting03/abstracts.html

John Skukalek, Grand Valley State University

Rings of Small Order

http://links.jstor.org/sici?sici=0025-5572(198003)2%3A64%3A427%3C9%3AROSO%3E2.0.CO%3B2-5

Rings of Small Order

Colin R. Fletcher

The Mathematical Gazette, Vol. 64, No. 427 (Mar., 1980), pp. 9-22

(Not even the abstract is available online without paying the

puiblisher, so I have no clue if this and the next reference are

relvant, but you might want to have a look)

http://portal.acm.org/citation.cfm?id=646656.700118

The Structure of Near-Rings of Small Order

http://home.wlu.edu/~dresdeng/smallrings/

http://www.algebra.uni-linz.ac.at/~noebsi/ringtable.html

Christof NÃ¶bauer

"Numbers of small rings"

"Transformation representations of minimal degree for small near-rings"

Benjamin Fine

Classification of Finite Rings of Order pÂ²

Mathematics Magazine, Vol. 66, No. 4

## Re: "order of a ring" with a vengeance

Thanks a lot for these links. I have added a couple of these which I was able to verify.

I found this source interesting as well as a little odd:

http://www.michmaa.org/meeting03/abstracts.html

John Skukalek, Grand Valley State University

Rings of Small Order

Part of the abstract reads:

A standard topic in a first class in group theory is the classification of all groups of small order. The classification of rings of small order is more difficult due to the more complicated structure. In this session we will present a complete classification scheme for all rings of order 5 or less.

Part of my motivation for researching cyclic rings was that ring classification is typically not studied in a first course in abstract algebra, whereas group classification and field classification is. The thing that weirds me out is that this person only considers classification schemes for rings up to order 5. If you are going to investigate rings of order 4 (which can be difficult, but I did manage to prove that there are 11 of them), why not do the easy cases of order 6 and order 7 as well?

Also, after looking at this source:

http://home.wlu.edu/~dresdeng/smallrings/

I contacted Dr. Dresden. He said that he will read Cyclic Rings and let me know what he thinks.