Let . An equivalence relation, called congruence (http://planetmath.org/Congruent2), can be defined on by iff divides . Note first of all that iff . Thus, without loss of generality, only nonnegative need be considered. Secondly, note that the case is not very interesting. If , then divides , which occurs exactly when . In this case, the set of all equivalence classes can be identified with . Thus, only positive need be considered. The set of all equivalence classes of under the given equivalence relation is called .
Some mathematicians consider the notation to be archaic and somewhat confusing. This matter of notation is most considerable when for some prime (http://planetmath.org/PrimeNumber) , as is used to refer to the -adic integers (http://planetmath.org/MathbbZ_p). To avoid this confusion, some mathematicians use the notation instead of . On the other hand, the notation should not cause confusion when is not prime, and is an intuitive shorthand way to write . Thus, others use when for some prime and otherwise. (The explanation of the usage of will come later.) Still others, especially those who are unfamiliar with the , use the notation exclusively. (In this entry, the notation is used exclusively, though it is highly recommended to use another notation when for some prime .)
One usually identifies an element of (which is technically a class (http://planetmath.org/EquivalenceClass), ) with the unique element in the class such that . One can use the division algorithm to establish that, for each class, an as described exists uniquely. (The set of all ’s as described is an example of a residue system.) Thus, the sets are finite with exactly elements. Addition and multiplication operations can also be defined on in a natural way that corresponds to the operations on . Under these operations, is a commutative ring with identity (http://planetmath.org/MultiplicativeIdentity) as well as a cyclic ring with behavior . When for some prime , is a field. In this case, the notation highlights the fact that the is a field. When is composite, has zero divisors and thus is neither a field nor an integral domain. Also note that is a zero ring, since all integers are equivalent (http://planetmath.org/Equivalent), yielding only one equivalence class.
The in both and is called the modulus. Performing computations such as addition, subtraction, multiplication, and taking exponents (http://planetmath.org/Exponent2) in one of the rings is called modular arithmetic.
|Date of creation||2013-03-22 15:58:10|
|Last modified on||2013-03-22 15:58:10|
|Last modified by||Wkbj79 (1863)|
|Synonym||integers mod n|