How Do You Know if a Ring Has a Multiplicative Identity

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Band

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A band in the mathematical sense is a set together with two binary operators and (commonly interpreted as add-on and multiplication, respectively) satisfying the following conditions:

1. Additive associativity: For all , ,

2. Additive commutativity: For all , ,

3. Additive identity: At that place exists an element such that for all , ,

4. Additive inverse: For every there exists such that ,

5. Left and right distributivity: For all , and ,

6. Multiplicative associativity: For all , (a band satisfying this property is sometimes explicitly termed an associative ring).

Weather 1-v are e'er required. Though non-associative rings exist, virtually all texts as well require condition vi (Itô 1986, pp. 1369-1372; p. 418; Zwillinger 1995, pp. 141-143; Harris and Stocker 1998; Knuth 1998; Korn and Korn 2000; Bronshtein and Semendyayev 2004).

Rings may also satisfy various optional conditions:

seven. Multiplicative commutativity: For all , (a ring satisfying this property is termed a commutative ring),

8. Multiplicative identity: There exists an element such that for all , (a band satisfying this property is termed a unit ring, or sometimes a "ring with identity"),

9. Multiplicative inverse: For each in , there exists an element such that for all , , where i is the identity element.

A band satisfying all additional properties half dozen-9 is chosen a field, whereas one satisfying just additional properties 6, 8, and ix is chosen a segmentation algebra (or skew field).

Some authors depart from the normal convention and require (under their definition) a ring to include additional properties. For example, Birkhoff and Mac Lane (1996) define a ring to take a multiplicative identity (i.due east., property 8).

Hither are a number of examples of rings defective particular conditions:

i. Without multiplicative associativity (sometimes also called nonassociative algebras): octonions, OEIS A037292,

2. Without multiplicative commutativity: Real-valued matrices, quaternions,

3. Without multiplicative identity: Fifty-fifty-valued integers,

4. Without multiplicative inverse: integers.

The word ring is brusk for the German word 'Zahlring' (number band). The French discussion for a ring is anneau, and the modern German word is Band, both pregnant (non then surprisingly) "ring." Fraenkel (1914) gave the first abstract definition of the ring, although this piece of work did not accept much impact. The term was introduced by Hilbert to draw rings like

By successively multiplying the new element , information technology eventually loops effectually to become something already generated, something like a band, that is, is new but is an integer. All algebraic numbers accept this property, e.g., satisfies .

A ring must contain at least one element, merely need not contain a multiplicative identity or be commutative. The number of finite rings of elements for , ii, ..., are 1, 2, 2, 11, 2, 4, 2, 52, 11, 4, 2, 22, two, four, 4, ... (OEIS A027623 and A037234; Fletcher 1980). If and are prime, in that location are two rings of size , 4 rings of size , 11 rings of size (Singmaster 1964, Dresden), 22 rings of size , 52 rings of size for , and 53 rings of size for (Ballieu 1947, Gilmer and Mott 1973; Dresden).

A ring that is commutative under multiplication, has a unit element, and has no divisors of zero is called an integral domain. A ring whose nonzero elements form a commutative multiplication group is chosen a field. The simplest rings are the integers , polynomials and in ane and ii variables, and square real matrices.

Rings which have been investigated and found to exist of involvement are normally named after one or more of their investigators. This do unfortunately leads to names which give very little insight into the relevant properties of the associated rings.

Renteln and Dundes (2005) give the following (bad) mathematical joke about rings:

Q: What'southward an Abelian group under improver, closed, associative, distributive, and bears a curse? A: The Ring of the Nibelung.

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See also

Abelian Group, Artinian Ring, Chow Band, Dedekind Band, Partitioning Algebra, Endomorphism Band, Field, Gorenstein Ring, Grouping, Group Band, Ideal, Integral Domain, Module, Nilpotent Element, Noetherian Ring, Noncommutative Ring, Number Field, Prime Ring, Prüfer Ring, Quotient Ring, Regular Ring, Ring of Integers, Ringoid, Semiprime Ring, Semiring, Semisimple Ring, Simple Ring, Trivial Band, Unit Ring, Zero Divisor Explore this topic in the MathWorld classroom

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References

Allenby, R. B. Rings, Fields, and Groups: An Introduction to Abstract Algebra, 2d ed. Oxford, England: Oxford University Press, 1991. Ballieu, R. "Anneaux finis; systèmes hypercomplexes de rang trois sur un corps commutatif." Ann. Soc. Sci. Bruxelles. Sér. I 61, 222-227, 1947. Beachy, J. A. Introductory Lectures on Rings and Modules. Cambridge, England: Cambridge University Press, 1999. Berrick, A. J. and Keating, One thousand. Eastward. An Introduction to Rings and Modules with G-Theory in View. Cambridge, England: Cambridge University Press, 2000. Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, fifth ed. New York: Macmillian, 1996. Bronshtein, I. N.; Semendyayev, K. A.; Musiol, One thousand.; and Muehlig, H. Handbook of Mathematics, fourth ed. New York: Springer-Verlag, 2004. Dresden, G. "Small Rings." http://domicile.wlu.edu/~dresdeng/smallrings/. Ellis, Chiliad. Rings and Fields. Oxford, England: Oxford University Press, 1993. Fine, B. "Nomenclature of Finite Rings of Club ." Math. Mag. 66, 248-252, 1993. Fletcher, C. R. "Rings of Pocket-size Order." Math. Gaz. 64, 9-22, 1980. Fraenkel, A. "Über dice Teiler der Null und dice Zerlegung von Ringen." J. reine angew. Math. 145, 139-176, 1914. Gilmer, R. and Mott, J. "Associative Rings of Order ." Proc. Japan Acad. 49, 795-799, 1973. Harris, J. Westward. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, 1998. Itô, One thousand. (Ed.). "Rings." §368 in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, 1986. Kleiner, I. "The Genesis of the Abstract Ring Concept." Amer. Math. Monthly 103, 417-424, 1996. Knuth, D. Eastward. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, 1998. Korn, Chiliad. A. and Korn, T. M. Mathematical Handbook for Scientists and Engineers. New York: Dover, 2000. Nagell, T. "Moduls, Rings, and Fields." §6 in Introduction to Number Theory. New York: Wiley, pp. 19-21, 1951. Renteln, P. and Dundes, A. "Foolproof: A Sampling of Mathematical Folk Humor." Notices Amer. Math. Soc. 52, 24-34, 2005. Singmaster, D. and Bloom, D. M. "Problem E1648." Amer. Math. Monthly 71, 918-920, 1964. Sloane, N. J. A. Sequences A027623 and A037234 in "The On-Line Encyclopedia of Integer Sequences." van der Waerden, B. L. A History of Algebra. New York: Springer-Verlag, 1985. Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002. Zwillinger, D. (Ed.). "Rings." §2.half dozen.3 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Printing, pp. 141-143, 1995.

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