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A certain kind of subring of the [[Group algebra|group algebra]]  $  \mathbf Z [ G] $
of a group  $  G $.

Let  $  G $
be a [[Finite group|finite group]] and  $  \pi = ( D _ {1} \dots D _ {n} ) $
a partition of  $  G $. 
For each  $  D \subset  G $, 
let  $  \overline{D}\; = \sum _ {g \in D }  g $
and  $  D  ^ {-1} = \{ {g  ^ {-1} } : {g \in D } \} $. 
Suppose that for each  $  D \in \pi $, 
$  D  ^ {-1} \in \pi $, 
and for all  $  D _ {i} , D _ {j} \in \pi $, 
$  \overline{D}\; _ {i} \overline{D}\; _ {j} = \sum _ {k=1} ^ {n} c _ {ij}  ^ {k} \overline{D}\; _ {k} $
for certain  $  c _ {ij}  ^ {k} \in \mathbf Z $. 
Then the  $  \overline{D}\; _ {1} \dots \overline{D}\; _ {n} $
form the basis (over  $  \mathbf Z $) 
of a subring of  $  \mathbf Z [ G] $. 
These subrings are called Schur rings. A unitary Schur ring is one which contains the unit element of  $  \mathbf Z [ G] $.

A subring  $  S $
of  $  \mathbf Z [ G] $
is a Schur ring over  $  G $
if and only if  $  x  ^ {(-1 )} \in S $
for all  $  x \in S $(
where  $  x  ^ {(- 1)} = \sum a _ {g} g  ^ {-1} $
if  $  x = \sum a _ {g} g $) 
and it is closed under the Hadamard product  $  ( \sum a _ {g} g ) ( \sum b _ {g} g ) = \sum a _ {g} b _ {g} g $.

A symmetric Schur ring  $  S $
is a Schur ring for which  $  x  ^ {(-1)} = x $
for all  $  x \in S $.

Historically, Schur rings were first studied by I. Schur [[#References|[a1]]] and H. Wielandt [[#References|[a2]]], who coined the name, in connection with the study of permutation groups; cf. [[#References|[a3]]]–[[#References|[a5]]] for applications of Schur rings to group theory. More recently it was discovered that they are also related to certain combinatorial structures, such as association schemes and strongly regular graphs, [[#References|[a6]]], [[#References|[a7]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Schur,   "Zur Theorie der einfach transitiven Permutationsgruppen"  ''Sitzungsber. Preuss. Akad. Wissenschaft. Berlin. Phys.-Math. Kl.''  (1933)  pp. 598–623</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Wielandt,   "Zur Theorie der einfach transitiven Permutationsgruppen II"  ''Math. Z.'' , '''52'''  (1949)  pp. 384–393</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O. Tamaschke,   "Schur-Ringe" , B.I. Wissenschaftsverlag Mannheim  (1970)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W.R. Scott,   "Group theory" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Wielandt,   "Finite permutation groups" , Acad. Press  (1964)  (Translated from German)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E. Bannai,   T. Ito,   "Algebraic combinatorics I: Association schemes" , Benjamin/Cummings  (1984)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S.L. Ma,   "On association schemes, Schur rings, strongly regular graphs and partial difference sets"  ''Ars Comb.'' , '''27'''  (1989)  pp. 211–220</TD></TR></table>