In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, a group ring is a
free module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a fiel ...
and at the same time a
ring, constructed in a natural way from any given ring and any given
group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.
If the ring is commutative then the group ring is also referred to as a group algebra, for it is indeed an
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
over the given ring. A group algebra over a field has a further structure of a
Hopf algebra Hopf is a German surname. Notable people with the surname include:
* Eberhard Hopf (1902–1983), Austrian mathematician
* Hans Hopf (1916–1993), German tenor
*Heinz Hopf (1894–1971), German mathematician
* Heinz Hopf (actor) (1934–2001), Swe ...
; in this case, it is thus called a
group Hopf algebra.
The apparatus of group rings is especially useful in the theory of
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
s.
Definition
Let ''G'' be a group, written multiplicatively, and let ''R'' be a ring. The group ring of ''G'' over ''R'', which we will denote by ''R''
'G''(or simply ''RG''), is the set of mappings of
finite support
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalles ...
(f(g) is nonzero for only finitely many elements g), where the module scalar product ''αf'' of a scalar ''α'' in ''R'' and a mapping ''f'' is defined as the mapping
, and the module group sum of two mappings ''f'' and ''g'' is defined as the mapping
. To turn the additive group ''R''
'G''into a ring, we define the product of ''f'' and ''g'' to be the mapping
:
The summation is legitimate because ''f'' and ''g'' are of finite support, and the ring axioms are readily verified.
Some variations in the notation and terminology are in use. In particular, the mappings such as are sometimes written as what are called "formal linear combinations of elements of ''G'', with coefficients in ''R''":
:
or simply
:
where this doesn't cause confusion.
[Polcino & Sehgal (2002), p. 131.]
Note that if the ring ''R'' is in fact a field ''K'', then the module structure of the group ring ''RG'' is in fact a vector space over ''K''.
Examples
1. Let , the
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order 3, with generator
and identity element 1
''G''. An element ''r'' of C
'G''can be written as
:
where ''z''
0, ''z''
1 and ''z''
2 are in C, the
complex numbers. This is the same thing as a
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
in variable
such that
i.e. C
'G''is isomorphic to the ring C
a">math>a.
Writing a different element ''s'' as
, their sum is
:
and their product is
:
Notice that the identity element 1
''G'' of ''G'' induces a canonical embedding of the coefficient ring (in this case C) into C
'G'' however strictly speaking the multiplicative identity element of C
'G''is 1⋅1
''G'' where the first ''1'' comes from C and the second from ''G''. The additive identity element is zero.
When ''G'' is a non-commutative group, one must be careful to preserve the order of the group elements (and not accidentally commute them) when multiplying the terms.
2. A different example is that of the
Laurent polynomials over a ring ''R'': these are nothing more or less than the group ring of the
infinite cyclic group Z over ''R''.
3. Let ''Q'' be the
quaternion group with elements
. Consider the group ring R''Q'', where R is the set of real numbers. An arbitrary element of this group ring is of the form
:
where
is a real number.
Multiplication, as in any other group ring, is defined based on the group operation. For example,
:
Note that R''Q'' is not the same as the skew field of
quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
over R. This is because the skew field of quaternions satisfies additional relations in the ring, such as
, whereas in the group ring R''Q'',
is not equal to
. To be more specific, the group ring R''Q'' has dimension 8 as a real
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, while the skew field of quaternions has dimension 4 as a
real vector space.
4. Another example of a non-abelian group ring is