ℤn
Let n∈ℤ. An equivalence relation, called congruence
(http://planetmath.org/Congruent2), can be defined on ℤ by a≡bmodn iff n divides b-a. Note first of all that a≡bmodn iff a≡bmod|n|. Thus, without loss of generality, only nonnegative n need be considered. Secondly, note that the case n=0 is not very interesting. If a≡bmod0, then 0 divides b-a, which occurs exactly when a=b. In this case, the set of all equivalence classes
can be identified with ℤ. Thus, only positive n need be considered. The set of all equivalence classes of ℤ under the given equivalence relation is called ℤn.
Some mathematicians consider the notation ℤn 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 ℤp is used to refer to the p-adic integers (http://planetmath.org/MathbbZ_p). To avoid this confusion, some mathematicians use the notation ℤ/nℤ instead of ℤn. On the other hand, the notation ℤn should not cause confusion when n is not prime, and is an intuitive shorthand way to write ℤ/nℤ. Thus, others use 𝔽p when n=p for some prime p and ℤn otherwise. (The explanation of the usage of 𝔽p will come later.) Still others, especially those who are unfamiliar with the , use the notation ℤn exclusively. (In this entry, the notation ℤn 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 ℤn (which is technically a class (http://planetmath.org/EquivalenceClass), ) with the unique element r in the class such that 0≤r<n. One can use the division algorithm 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 ℤn are finite with exactly n elements. Addition
and multiplication operations
can also be defined on ℤn in a natural way that corresponds to the operations on ℤ. Under these operations, ℤn 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, ℤn is a field. In this case, the notation 𝔽p highlights the fact that the is a field. When n is composite, ℤn has zero divisors
and thus is neither a field nor an integral domain
. Also note that ℤ1 is a zero ring
, since all integers are equivalent
(http://planetmath.org/Equivalent), yielding only one equivalence class.
The n in both ℤn and a≡bmodn is called the modulus. Performing computations such as addition, subtraction, multiplication, and taking exponents (http://planetmath.org/Exponent2) in one of the rings ℤn is called modular arithmetic.
Title | ℤn |
---|---|
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 |