Définitions
1.NounGiven ring R with identity not equal to zero, and group
G
=
{
g
1
,
g
2
,
.
.
.
,
g
n
}
{\displaystyle G=\{g_{1},g_{2},...,g_{n}\}}
, the group ring RG has elements of the form
a
1
g
1
+
a
2
g
2
+
.
.
.
+
a
n
g
n
{\displaystyle a_{1}g_{1}+a_{2}g_{2}+...+a_{n}g_{n}}
(where
a
i
∈
R
{\displaystyle a_{i}\in R}
) such that the sum of
a
1
g
1
+
a
2
g
2
+
.
.
.
+
a
n
g
n
{\displaystyle a_{1}g_{1}+a_{2}g_{2}+...+a_{n}g_{n}}
and
b
1
g
1
+
b
2
g
2
+
.
.
.
+
b
n
g
n
{\displaystyle b_{1}g_{1}+b_{2}g_{2}+...+b_{n}g_{n}}
is
(
a
1
+
b
1
)
g
1
+
(
a
2
+
b
2
)
g
2
+
.
.
.
+
(
a
n
+
b
n
)
g
n
{\displaystyle (a_{1}+b_{1})g_{1}+(a_{2}+b_{2})g_{2}+...+(a_{n}+b_{n})g_{n}}
and the product is
∑
k
=
1
n
(
∑
g
i
g
j
=
g
k
a
i
b
j
)
g
k
{\displaystyle \sum _{k=1}^{n}\left(\sum _{g_{i}g_{j}=g_{k}}a_{i}b_{j}\right)g_{k}}
.