# 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. 1.

the zero ring,

2. 2.

the ring of integers $\mathbb{Z}$,

3. 3.

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

4. 4.

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

5. 5.

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

6. 6.

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

7. 7.

other rings of rational numbers

8. 8.

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

9. 9.

the rational numbers $\mathbb{Q}$,

10. 10.

the real numbers $\mathbb{R}$,

11. 11.
12. 12.

the complex numbers $\mathbb{C}$,

13. 13.

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

## Examples of non-commutative rings

1. 1.

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

2. 2.

the set of square matrices $M_{n}(R)$, with $n>1$,

3. 3.

the set of triangular matrices (upper or lower, but not both in the same set),

4. 4.

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

5. 5.
6. 6.

Let $A$ be an abelian group. Then the set of group endomorphisms $f:A\to A$ forms a ring $\operatorname{End}A$, 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 $f\circ(g+h)\neq f\circ g+f\circ g$ and so this set forms only a near ring.

## Change of rings (rings generated from other rings)

Let $R$ be a ring.

1. 1.

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

2. 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. 3.

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

4. 4.

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

5. 5.

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

6. 6.

$M_{n\times n}(R)$ is the $n\times n$ matrix ring over $R$.

7. 7.

A special case of Example 6 under the section on non-commutative rings is the ring of endomorphisms over a ring $R$.

8. 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. 9.

For any non-empty 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 $\bigoplus_{M}R$. For instance, if $M=\{1,2\}$, then $R^{M}\cong R\oplus R$.

10. 10.

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

11. 11.

Let $S,T$ be subrings of $R$. Then

 $\begin{pmatrix}S&R\\ 0&T\end{pmatrix}:=\Big{\{}\begin{pmatrix}s&r\\ 0&t\end{pmatrix}\mid r\in R,s\in S,t\in T\Big{\}}$

with the usual matrix addition and multiplication is a ring.

Title examples of rings ExamplesOfRings 2013-03-22 15:00:42 2013-03-22 15:00:42 matte (1858) matte (1858) 42 matte (1858) Example msc 16-00 msc 13-00 CommutativeRing Ring