examples of rings
Rings in this article are assumed to have a commutative^{} addition^{} with negatives and an associative multiplication. However, it is not generally assumed that all rings included here are unital.
Examples of commutative rings

1.
the zero ring^{},

2.
the ring of integers^{} $\mathbb{Z}$,

3.
the ring of even integers $2\mathbb{Z}$ (a ring without identity^{}), or more generally, $n\mathbb{Z}$ for any integer $n$,

4.
the integers modulo $n$ (http://planetmath.org/MathbbZ_n), $\mathbb{Z}/n\mathbb{Z}$,

5.
the ring of integers ${\mathcal{O}}_{K}$ of a number field^{} $K$,

6.
the $p$integral rational numbers (http://planetmath.org/PAdicValuation) (where $p$ is a prime number^{}),

7.
other rings of rational numbers

8.
the $p$adic integers (http://planetmath.org/PAdicIntegers) ${\mathbb{Z}}_{p}$ and the $p$adic numbers ${\mathbb{Q}}_{p}$,

9.
the rational numbers $\mathbb{Q}$,

10.
the real numbers $\mathbb{R}$,

11.
rings and fields of algebraic numbers,

12.
the complex numbers^{} $\u2102$,

13.
The set ${2}^{A}$ of all subsets of a set $A$ is a ring. The addition is the symmetric difference^{} “$\mathrm{\u25b3}$” and the multiplication the set operation^{} intersection^{} “$\cap $”. Its additive identity is the empty set^{} $\mathrm{\varnothing}$, and its multiplicative identity^{} is the set $A$. This is an example of a Boolean ring^{}.
Examples of noncommutative rings

1.
the quaternions, $\mathbb{H}$, also known as the Hamiltonions. This is a finite dimensional division ring over the real numbers, but noncommutative.

2.
the set of square matrices^{} ${M}_{n}(R)$, with $n>1$,

3.
the set of triangular matrices^{} (upper or lower, but not both in the same set),

4.
strict triangular matrices (http://planetmath.org/StrictUpperTriangularMatrix) (same condition as above),
 5.

6.
Let $A$ be an abelian group. Then the set of group endomorphisms^{} $f:A\to A$ forms a ring $\mathrm{End}A$, with addition defined elementwise ($(f+g)(a)=f(a)+g(a)$) and multiplication the functional^{} composition. It is the ring of operators over $A$.
By contrast, the set of all functions $\{f:A\to A\}$ are closed to addition and composition, however, there are generally functions $f$ such that $f\circ (g+h)\ne f\circ g+f\circ g$ and so this set forms only a near ring.
Change of rings (rings generated from other rings)
Let $R$ be a ring.

1.
If $I$ is an ideal of $R$, then the quotient^{} $R/I$ is a ring, called a quotient ring^{}.

2.
$R[x]$ is the polynomial ring over $R$ in one indeterminate $x$ (or alternatively, one can think that $R[x]$ is any transcendental extension^{} ring of $R$, such as $\mathbb{Z}[\pi ]$ is over $\mathbb{Z}$),

3.
$R(x)$ is the field of rational functions in $x$,

4.
$R[[x]]$ is the ring of formal power series in $x$,

5.
$R((x))$ is the ring of formal Laurent series in $x$,

6.
${M}_{n\times n}(R)$ is the $n\times n$ matrix ring over $R$.

7.
A special case of Example 6 under the section^{} on noncommutative rings is the ring of endomorphisms over a ring $R$.

8.
For any group $G$, the group ring^{} $R[G]$ is the set of formal sums of elements of $G$ with coefficients in $R$.

9.
For any nonempty set $M$ and a ring $R$, the set ${R}^{M}$ of all functions from $M$ to $R$ may be made a ring $({R}^{M},+,\cdot )$ by setting for such functions $f$ and $g$
$$(f+g)(x):=f(x)+g(x),(fg)(x):=f(x)g(x)\forall x\in M.$$ This ring is the often denoted ${\oplus}_{M}R$. For instance, if $M=\{1,2\}$, then ${R}^{M}\cong R\oplus R$.

10.
If $R$ is commutative, the ring of fractions^{} ${S}^{1}R$ where $S$ is a multiplicative subset of $R$ not containing 0.

11.
Let $S,T$ be subrings of $R$. Then
$$\left(\begin{array}{cc}\hfill S\hfill & \hfill R\hfill \\ \hfill 0\hfill & \hfill T\hfill \end{array}\right):=\{\left(\begin{array}{cc}\hfill s\hfill & \hfill r\hfill \\ \hfill 0\hfill & \hfill t\hfill \end{array}\right)\mid r\in R,s\in S,t\in T\}$$ with the usual matrix addition^{} and multiplication is a ring.
Title  examples of rings 

Canonical name  ExamplesOfRings 
Date of creation  20130322 15:00:42 
Last modified on  20130322 15:00:42 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  42 
Author  matte (1858) 
Entry type  Example 
Classification  msc 1600 
Classification  msc 1300 
Related topic  CommutativeRing 
Related topic  Ring 