The group ring ![R[G]](/images/equations/GroupRing/Inline1.svg)
of a group 
over a commutative ring 
is the ring
of formal sums 
, where 
and only finitely many coefficients are nonzero.
Addition is coefficientwise, and multiplication
extends that of 
by distributivity, with elements
of 
commuting with elements
of 
. When 
is a field, this is the group
algebra (Sharifi, Definition 5.9.17). For example, ![Z[G]](/images/equations/GroupRing/Inline10.svg)
permits integer coefficients,
as used in the module formulation of Wall's
D2 problem (Johnson 2003, p. 19).
Group Ring
See also
CG, Graded Algebra, Group Algebra, Group-Ring Determinant Conjecture, Hopf Algebra, Laurent Polynomial, Ring, Semigroup RingExplore with Wolfram|Alpha
References
Johnson, F. E. A. "Stable Modules and Wall's
-Problem." Comment. Math.
Helv. 78, 18-44, 2003. https://doi.org/10.1007/S000140300001.Sharifi,
R. "Associative Algebras." §5.9 in Abstract Algebra. https://www.math.ucla.edu/~sharifi/notes/algebra-ch05.html.Referenced on Wolfram|Alpha
Group RingCite this as:
Weisstein, Eric W. "Group Ring." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GroupRing.html