PlanetMath: examples of rings

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examples of rings

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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.

the zero ring,
the ring of integers $\mathbb{Z}$ ,
the ring of even integers $2\mathbb{Z}$ (a ring without identity), or more generally, $n\mathbb{Z}$ for any integer ,
the integers modulo , $\mathbb{Z}/n\mathbb{Z}$ ,
the ring of integers $\mathcal{O}_K$ of a number field ,
the -integral rational numbers (where is a prime number),
other rings of rational numbers
the -adic integers $\mathbb{Z}_p$ and the -adic numbers $\mathbb{Q}_p$ ,
the rational numbers $\mathbb{Q}$ ,
the real numbers $\mathbb{R}$ ,
rings and fields of algebraic numbers,
the complex numbers $\mathbb{C}$ ,
The set $2^{A}$ of all subsets of a set 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 . This is an example of a Boolean ring.

the quaternions, $\mathbb{H}$ , also known as the Hamiltonions. This is a finite dimensional division ring over the real numbers, but noncommutative.
the set of square matrices , with ,
the set of triangular matrices (upper or lower, but not both in the same set),
strict triangular matrices (same condition as above),
Klein 4-ring,
Let be an abelian group. Then the set of group endomorphisms $f:A\to A$ forms a ring ${\mathrm{End}}A$ , with addition defined elementwise ( ) and multiplication the functional composition. It is the ring of operators over .
By contrast, the set of all functions $\{f:A\to A\}$ are closed to addition and composition, however, there are generally functions such that $f\circ(g+h)\neq f\circ g+f\circ g$ and so this set forms only a near ring.

Let

be a ring.

If is an ideal of , then the quotient is a ring, called a quotient ring.
is the polynomial ring over in one indeterminate (or alternatively, one can think that is any transcendental extension ring of , such as $\mathbb{Z}[\pi]$ is over $\mathbb{Z}$ ),
is the field of rational functions in ,
is the ring of formal power series in ,
is the ring of formal Laurent series in ,
$M_{n\times n}(R)$ is the $n\times n$ matrix ring over .
A special case of Example 6 under the section on non-commutative rings is the ring of endomorphisms over a ring .
For any group , the group ring is the set of formal sums of elements of with coefficients in .
For any non-empty set and a ring , the set of all functions from to may be made a ring $(R^M,\,+,\,\cdot)$ by setting for such functions and
$\displaystyle (f\!+\!g)(x) := f(x)+g(x), \,\,\, (fg)(x) := f(x)g(x)\,\,\, \forall x\in M.$
This ring is the often denoted $\bigoplus_{M} R$ . For instance, if $M=\{1,2\}$ , then $R^M\cong R\oplus R$ .
If is commutative, the ring of fractions $S^{-1}R$ where is a multiplicative subset of not containing 0.
Let be subrings of . Then
$\displaystyle \begin{pmatrix}S&R \\ 0&T \end{pmatrix}:=\Big\lbrace \begin{pmatrix}s&r \\ 0&t \end{pmatrix}\mid r\in R, s\in S, t\in T \Big\rbrace$
with the usual matrix addition and multiplication is a ring.