examples of rings (original) (raw)

Examples of commutative rings

4. 1. the integers modulo n (http://planetmath.org/MathbbZ\_n), ℤ/n⁢ℤ,
7. 1. other rings of rational numbers
8. 1. the p-adic integers (http://planetmath.org/PAdicIntegers) ℤp and the p-adic numbers ℚp,
9. 1. the rational numbers ℚ,

Examples of non-commutative rings

3. 1. the set of triangular matrices (upper or lower, but not both in the same set),
4. 1. strict triangular matrices (http://planetmath.org/StrictUpperTriangularMatrix) (same condition as above),
6. 1. 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)≠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.

2. 1. 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),
5. 1. R⁢((x)) is the ring of formal Laurent series in x,
7. 1. A special case of Example 6 under the section on non-commutative rings is the ring of endomorphisms over a ring R.
8. 1. For any group G, the group ring R⁢[G] is the set of formal sums of elements of G with coefficients in R.
9. 1. 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.
      This ring is the often denoted ⊕MR. For instance, if M={1,2}, then RM≅R⊕R.

11. 1. Let S,T be subrings of R. Then

       (SR0T):={(sr0t)∣r∈R,s∈S,t∈T}
       with the usual matrix addition and multiplication is a ring.