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In
ring theory, a branch of
abstract algebra, a '''ring''' is an
algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the
integers.
== History ==
See
Ring theory
== Definition and notation ==
A ring is an
abelian group (''R'', +), together with a second
binary operation * such that for all ''a'', ''b'' and ''c'' in ''R'',
: ''a'' * ''b'' is in ''R'' [Closed]
: ''a'' * (''b''*''c'') = (''a*b'') * ''c'' [Associativity]
: ''a'' * (''b''+''c'') = (''a*b'') + (''a*c'') [* Pre-Distributive over +]
: (''a''+''b'') * ''c'' = (''a''*''c'') + (''b''*''c'') [* Post-Distributive over +]
and such that there exists a ''multiplicative identity'', or ''unity'',
that is, an element 1 so that for all ''a'' in ''R'',
: ''a''*1 = 1*''a'' = ''a''
(Some authors omit the requirement for a multiplicative identity, and call those rings which do have multiplicative identities '''unitary rings''', '''
unital rings''' or '''rings with a 1'''. Similarly, the requirement for the ring multiplication to be
associative is sometimes dropped, and rings in which the associative law holds are called '''associative rings'''. In this encyclopedia, associativity and the existence of a multiplicative identity are taken to be part of the definition of a ring.)
Note that the
commutative law,
: ''a''*''b'' = ''b''*''a'' for all ''a'',''b'' ∈ ''R''
is not among the ring axioms listed above; rings that satisfy this law (such as the ring of integers) are called
commutative rings. Not all rings are commutative; see, for example, Matrix rings, described below.
The identity element with respect to + is called the zero element of the ring and written as 0. The symbol * is usually omitted from the notation, and the standard
order of operation rules are used, so that e.g. ''a''+''bc'' is an abbreviation for ''a''+(''b''*''c''). The additive inverse of the element ''x'' in a ring is written as -''x''.
In a ring we have 0=1 if and only if we are dealing with the ''trivial ring'' {0} with a single element.
An element ''a'' in a ring is called a '''unit''' if it is invertible with respect to multiplication, i.e., if there is an element ''b'' in the ring such that
: ''ab'' = ''ba'' = 1
If that is the case, then ''b'' is uniquely determined by ''a'' and we write ''a''
-1 = ''b''.
== Examples ==
* The motivating example is the ring of
integers with the two operations of addition and multiplication. This is a commutative ring.
* The
rational,
real and
complex numbers form rings, in fact they are even
fields. These are likewise commutative rings.
* More generally, every field is a commutative ring.
* If ''n'' is a positive integer, then the set '''Z'''/''n'''''Z''' of integers modulo ''n'' forms a ring with ''n'' elements (see
modular arithmetic).
* The
split-complex plane D is a ring useful in modern physics.
* The set of all
continuous real-valued
functions defined on the
interval [''a'', ''b''] forms a ring (even an
associative algebra). The operations are addition and multiplication of functions.
* The set of all
polynomials over some common coefficient ring forms a ring.
* For any ring ''R'' and any natural number ''n'', the set of all square ''n''-by-''n''
matrices with entries from ''R'', forms a ring with matrix addition and matrix multiplication as operations. For ''n''=1, this matrix ring is just (isomorphic to) ''R'' itself. For ''n''>2, this matrix ring is an example of a noncommutative ring (unless ''R'' is the trivial ring).
* The ''trivial ring'' {0} has only one element which serves both as additive and multiplicative identity.
* If ''G'' is an abelian
group, then the
endomorphisms of ''G'' form a ring, the endomorphism ring End(''G'') of ''G''. The operations in this ring are addition and composition of endomorphisms.
* If ''S'' is a set, then the
power set of ''S'' becomes a ring if we define addition to be the
symmetric difference of sets and multiplication to be
intersection. This is an example of a
Boolean ring.
* The set of
formal power series ''R''[[''X''
1,...,''X''
''n'']] over a commutative ring ''R'' is a ring.
* The set of all functions in ''n'' complex variables
holomorphic at the origin is a ring.
* The
Weyl algebra over the
field ''k'' is generated by 2 elements ''x'' and ''y'' subject to the relation ''xy-yx=1''.
* If ''G'' is a
group and ''R'' is a ring the
group ring of ''G'' over ''R'' is a
free module over having ''G'' as basis. Multiplication is defined by the rules that the elements of ''G'' commute with the elements of ''R'' and multiply together as they do in the group ''G''.
* The
free algebra on a set of indeterminates over the ring ''R'' is a further example of a noncommutative ring provided there is more than one indeterminate.
* The set of endomorphisms of an object in an abelian category is a ring.
* The path algebra of a
quiver is another useful noncommutative ring.
== Simple theorems ==
From the axioms, one can immediately deduce that, for all elements ''a'' and ''b'' of a ring, we have
* 0''a'' = ''a''0 = 0
* (-1)''a'' = -''a''
* (-''a'')''b'' = ''a''(-''b'') = -(''ab'')
* (''ab'')
-1=''b''
-1 ''a''
-1 if both ''a'' and ''b'' are invertible, and hence the set of all invertible elements in a ring is closed under multiplication * and forms a
group, the ''group of units'' of the ring.
== Constructing new rings from given ones ==
*If a subset ''S'' of a ring (''R'',+,*) together with the operations + and * restricted on ''S'' is itself a ring, and the identity element 1 of ''R'' is contained in ''S'', then ''S'' is called a ''subring'' of (''R'',+,*).
*The centre of a ring ''R'' is the set of elements of ''R'' that commute with every element of ''R''; that is, ''c'' lies in the centre if ''cr''=''rc'' for every ''r'' in ''R''. The centre is a subring of ''R''. We say that a subring ''S'' of ''R'' is central if it is a subring of the centre of ''R''.
*The ''direct sum'' of two rings ''R'' and ''S'' is the
cartesian product ''R''×''S'' together with the operations
:(''r''
1, ''s''
1) + (''r''
2, ''s''
2) = (''r''
1+''r''
2, ''s''
1+''s''
2) and
:(''r''
1, ''s''
1) * (''r''
2, ''s''
2) = (''r''
1*''r''
2, ''s''
1*''s''
2).
*Given a ring ''R'' and an
ideal ''I'' of ''R'', the ''
quotient ring'' (or ''factor ring'') ''R''/''I'' is the set of cosets of ''I'' together with the operations
:(''a+I'') + (''b+I'') = (''a''+''b'') + ''I'' and
:(''a+I'') * (''b+I'') = (''a''*''b'') + ''I''.
*Since any ring is both a left and right
module over itself, it is possible to construct the
tensor product of ''R'' over a ring ''S'' with another ring ''T'' to get another ring provided ''S'' is a central subring of ''R'' and ''T''.
==Glossary and related topics==
See
Glossary of ring theory for more definitions in
ring theory.
== See also ==
*
wikibooks:Abstract algebra:Rings
de:Ringtheorie
es:Anillo (matemticas)
eo:Ringo (Algebro)
fr:Anneau
it:Anello (matematica)
nl:Ring (wiskunde)
ja:環論
pl:Pierścień (matematyka)
pt:Anel
ru:Кольцо (алгебра)
sl:Kolobar
zh:环
