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examples of rings∗
matte†
2013-03-21 18:40:43
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 Z,
3. the ring of even integers 2Z (a ring without identity), or more generally,
nZ for any integer n,
4. the integers modulo n, Z/nZ,
5. the ring of integers OK of a number field K,
6. the p-integral rational numbers (where p is a prime number),
7. other rings of rational numbers
8. the p-adic integers Zp and the p-adic numbers Qp ,
9. the rational numbers Q,
10. the real numbers R,
11. rings and fields of algebraic numbers,
12. the complex numbers C,
13. The set 2A of all subsets of a set A is a ring. The addition is the symmetric
difference “4” and the multiplication the set operation intersection “∩”.
Its additive identity is the empty set ∅, and its multiplicative identity is
the set A. This is an example of a Boolean ring.
∗ hExamplesOfRingsi
created: h2013-03-21i by: hmattei version: h36718i Privacy setting:
h1i hExamplei h16-00i h13-00i
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Examples of non-commutative rings
1. the quaternions, 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 Mn (R), with n > 1,
3. the set of triangular matrices (upper or lower, but not both in the same
set),
4. strict triangular matrices (same condition as above),
5. Klein 4-ring,
6. Let A be an abelian group. Then the set of group endomorphisms f : A →
A forms a ring 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 → A} are closed to addition and
composition, however, there are generally functions f such that f ◦(g+h) 6=
f ◦ g + f ◦ 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
Z[π] is over 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. Mn×n (R) is the n × 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 RM of all functions from
M to R may be made a ring (RM , +, ·) by setting for such functions f
and g
(f +g)(x) := f (x) + g(x), (f g)(x) := f (x)g(x) ∀x ∈ M.
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This ring is the often denoted
RM ∼
= R ⊕ R.
L
M
R. For instance, if M = {1, 2}, then
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
n s r o
S R
:=
| r ∈ R, s ∈ S, t ∈ T
0 T
0 t
with the usual matrix addition and multiplication is a ring.
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