Editing Ring (mathematics) (section)

== Basic examples ==
{{see also|Associative algebra#Examples}}

=== Commutative rings ===
* The prototypical example is the ring of integers with the two operations of addition and multiplication.
* The rational, real and complex numbers are commutative rings of a type called [[field (mathematics)|fields]].
* A unital associative [[algebra over a ring|algebra over a commutative ring]] {{mvar|R}} is itself a ring as well as an [[module (mathematics)|{{mvar|R}}-module]]. Some examples:
** The algebra {{math|''R''[''X'']}} of [[polynomial ring|polynomials]] with coefficients in {{mvar|R}}.
** The algebra <math>R[[X_1, \dots, X_n]]</math> of [[formal power series]] with coefficients in {{mvar|R}}.
** The set of all [[continuous function|continuous]] real-valued [[function (mathematics)|functions]] defined on the real line forms a commutative {{tmath|\R}}-algebra. The operations are [[pointwise]] addition and multiplication of functions.
** Let {{mvar|X}} be a set, and let {{mvar|R}} be a ring. Then the set of all functions from {{mvar|X}} to {{mvar|R}} forms a ring, which is commutative if {{mvar|R}} is commutative.
<!--* <math>\Z [c],</math> the integers with a real or complex number {{mvar|c}} adjoined. As a {{tmath|\Z}}-module, it is free of infinite rank if {{mvar|c}} is [[transcendental number|transcendental]], free of finite rank if {{mvar|c}} is an algebraic integer, and not free otherwise.
* <math>\Z [1/10],</math> the set of [[decimal fraction]]s.  Not free as a {{math|\Z}}-module.-->
* The ring of [[quadratic integers]], the integral closure of {{tmath|\Z}} in a quadratic extension of {{tmath|\Q.}} It is a subring of the ring of all [[algebraic integers]].
<!--* <math>\Z [i],</math> the [[Gaussian integer]]s.
* <math>\Z [\left(1 + \sqrt{-3}\right)/2],</math> the [[Eisenstein integer]]s. 
* The previous two examples are the cases {{math|1=''n'' = 4}} and {{math|1=''n'' = 3}} of the [[cyclotomic field|cyclotomic ring]] {{tmath|\Z[\zeta_n].}}
* The previous four examples are cases of the [[ring of integers]] of a [[number field]] {{mvar|K}}, defined as the set of [[algebraic integer]]s in {{mvar|K}}.-->
<!--* The set of all algebraic integers in {{tmath|\C}} forms a ring called the [[integral closure]] of {{tmath|\Z}} in {{tmath|\C.}}-->
* The ring of [[profinite integer]]s {{tmath|\widehat\Z,}} the (infinite) product of the rings of {{mvar|p}}-adic integers {{tmath|\Z _p}} over all prime numbers {{mvar|p}}.
* The [[Hecke algebra|Hecke ring]], the ring generated by Hecke operators.
* If {{mvar|S}} is a set, then the [[power set]] of {{mvar|S}} becomes a ring if we define addition to be the [[symmetric difference]] of sets and multiplication to be [[intersection (set theory)|intersection]]. This is an example of a [[Boolean ring]].

=== Noncommutative rings ===
* For any ring {{mvar|R}} and any natural number {{mvar|n}}, the set of all square {{mvar|n}}-by-{{mvar|n}} [[matrix (mathematics)|matrices]] with entries from {{mvar|R}}, forms a ring with matrix addition and matrix multiplication as operations. For {{math|1=''n'' = 1}}, this matrix ring is isomorphic to {{mvar|R}} itself. For {{math|''n'' > 1}} (and {{mvar|R}} not the zero ring), this matrix ring is noncommutative.
* If {{math|''G''}} is an [[abelian group]], then the [[group homomorphism|endomorphisms]] of {{math|''G''}} form a ring, the [[endomorphism ring]] {{math|End(''G'')}} of&nbsp;{{math|''G''}}. The operations in this ring are addition and composition of endomorphisms. More generally, if {{mvar|V}} is a [[left module]] over a ring {{mvar|R}}, then the set of all {{mvar|R}}-linear maps forms a ring, also called the endomorphism ring and denoted by {{math|End{{sub|''R''}}(''V'')}}.
*The [[endomorphism ring of an elliptic curve]]. It is a commutative ring if the elliptic curve is defined over a field of characteristic zero.
* If {{math|''G''}} is a [[group (mathematics)|group]] and {{mvar|R}} is a ring, the [[group ring]] of {{math|''G''}} over {{mvar|R}} is a [[free module]] over {{mvar|R}} having {{math|''G''}} as basis. Multiplication is defined by the rules that the elements of {{math|''G''}} commute with the elements of {{mvar|R}} and multiply together as they do in the group {{math|''G''}}.
* The [[ring of differential operators]] (depending on the context). In fact, many rings that appear in analysis are noncommutative. For example, most [[Banach algebra]]s are noncommutative.

=== Non-rings ===
* The set of [[natural number]]s {{tmath|\N}} with the usual operations is not a ring, since {{tmath|(\N, +)}} is not even a [[group (mathematics)|group]] (not all the elements are [[inverse element|invertible]] with respect to addition – for instance, there is no natural number which can be added to {{math|3}} to get {{math|0}} as a result). There is a natural way to enlarge it to a ring, by including negative numbers to produce the ring of integers {{tmath|\Z.}} The natural numbers (including {{math|0}}) form an algebraic structure known as a [[semiring]] (which has all of the axioms of a ring excluding that of an additive inverse).
* Let {{mvar|R}} be the set of all continuous functions on the real line that vanish outside a bounded interval that depends on the function, with addition as usual but with multiplication defined as [[convolution]]: <math display="block">(f * g)(x) = \int_{-\infty}^\infty f(y)g(x - y) \, dy.</math> Then {{mvar|R}} is a rng, but not a ring: the [[Dirac delta function]] has the property of a multiplicative identity, but it is not a function and hence is not an element of&nbsp;{{mvar|R}}.