Let n. An equivalence relationMathworldPlanetmath, called congruenceMathworldPlanetmathPlanetmathPlanetmath (http://planetmath.org/Congruent2), can be defined on by abmodn iff n divides b-a. Note first of all that abmodn iff abmod|n|. Thus, without loss of generality, only nonnegative n need be considered. Secondly, note that the case n=0 is not very interesting. If abmod0, then 0 divides b-a, which occurs exactly when a=b. In this case, the set of all equivalence classesMathworldPlanetmath can be identified with . Thus, only positive n need be considered. The set of all equivalence classes of under the given equivalence relation is called n.

Some mathematicians consider the notation n to be archaic and somewhat confusing. This matter of notation is most considerable when n=p for some prime (http://planetmath.org/PrimeNumber) p, as p is used to refer to the p-adic integers (http://planetmath.org/MathbbZ_p). To avoid this confusion, some mathematicians use the notation /n instead of n. On the other hand, the notation n should not cause confusion when n is not prime, and is an intuitive shorthand way to write /n. Thus, others use 𝔽p when n=p for some prime p and n otherwise. (The explanation of the usage of 𝔽p will come later.) Still others, especially those who are unfamiliar with the , use the notation n exclusively. (In this entry, the notation n is used exclusively, though it is highly recommended to use another notation when n=p for some prime p.)

One usually identifies an element of n (which is technically a class (http://planetmath.org/EquivalenceClass), ) with the unique element r in the class such that 0r<n. One can use the division algorithmPlanetmathPlanetmath to establish that, for each class, an r as described exists uniquely. (The set of all r’s as described is an example of a residue system.) Thus, the sets n are finite with exactly n elements. AdditionPlanetmathPlanetmath and multiplication operationsMathworldPlanetmath can also be defined on n in a natural way that corresponds to the operations on . Under these operations, n is a commutative ring with identity (http://planetmath.org/MultiplicativeIdentity) as well as a cyclic ring with behavior 1. When n=p for some prime p, n is a field. In this case, the notation 𝔽p highlights the fact that the is a field. When n is composite, n has zero divisorsMathworldPlanetmath and thus is neither a field nor an integral domainMathworldPlanetmath. Also note that 1 is a zero ringMathworldPlanetmath, since all integers are equivalentMathworldPlanetmathPlanetmathPlanetmath (http://planetmath.org/Equivalent), yielding only one equivalence class.

The n in both n and abmodn is called the modulusMathworldPlanetmathPlanetmath. Performing computations such as addition, subtraction, multiplication, and taking exponents (http://planetmath.org/Exponent2) in one of the rings n is called modular arithmetic.

Title n
Canonical name mathbbZn
Date of creation 2013-03-22 15:58:10
Last modified on 2013-03-22 15:58:10
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 25
Author Wkbj79 (1863)
Entry type Definition
Classification msc 13-00
Classification msc 13M05
Classification msc 11-00
Synonym integers mod n
Related topic ResidueSystems
Related topic MathbbZ
Related topic CyclicRingsThatAreIsomorphicToKmathbbZ_kn
Related topic Congruences
Related topic EquivalenceRelation
Defines modulus
Defines modular arithmetic