examples of rings
Examples of commutative rings
the zero ring,
the ring of integers ,
the integers modulo (http://planetmath.org/MathbbZ_n), ,
the ring of integers of a number field ,
other rings of rational numbers
the -adic integers (http://planetmath.org/PAdicIntegers) and the -adic numbers ,
the rational numbers ,
the real numbers ,
rings and fields of algebraic numbers,
the complex numbers ,
Examples of non-commutative rings
the set of square matrices , with ,
the set of triangular matrices (upper or lower, but not both in the same set),
strict triangular matrices (http://planetmath.org/StrictUpperTriangularMatrix) (same condition as above),
By contrast, the set of all functions are closed to addition and composition, however, there are generally functions such that and so this set forms only a near ring.
Change of rings (rings generated from other rings)
Let be a ring.
is the field of rational functions in ,
is the ring of formal power series in ,
is the ring of formal Laurent series in ,
is the matrix ring over .
For any non-empty set and a ring , the set of all functions from to may be made a ring by setting for such functions and
This ring is the often denoted . For instance, if , then .
|Title||examples of rings|
|Date of creation||2013-03-22 15:00:42|
|Last modified on||2013-03-22 15:00:42|
|Last modified by||matte (1858)|