examples of rings

Rings in this article are assumed to have a commutative additionwith negatives and an associative multiplication. However, itis 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 ${𝒪}_K$ 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 $ℝ$,

11. rings and fields of algebraic numbers,

12. the complex numbers $ℂ$,

13. The set $2^A$ of all subsets of a set $A$ is a ring. The addition is the symmetric difference “$\mathrm{△}$” 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 non-commutative rings

1. the quaternions, $ℍ$, also known as the Hamiltonions. This is a finite dimensional division ringover 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. Klein 4-ring,

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 setforms 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 non-commutative 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 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 ${\oplus }_{M}R$. 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

    with the usual matrix addition and multiplication is a ring.