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A ring the additive group of which is the graded cohomology group

$$ 
\oplus_{n=0}^  \infty  
H  ^ {n} ( X , A ) ,
$$

where  $  X $
is a chain complex,  $  A $
is a coefficient group and the multiplication is defined by the linear set of mappings

$$ 
\nu _ {m,n} :  H  ^ {m} ( X , A ) \otimes H  ^ {n} ( X , A )  \rightarrow  H  ^ {m+n}
( X , A ) ,
$$

for all  $  m , n \geq  0 $, 
which are the inner cohomology multiplications (cup products). The cohomology ring turns out to be equipped with the structure of a graded ring.

For the existence of the mappings  $  \nu _ {m,n} $
it is enough to have a set of mappings  $  \widehat \nu   _ {m,n} :  X _ {m+n} \rightarrow X _ {m} \otimes X _ {n} $
satisfying certain additional properties, and a mapping  $  A \otimes A \rightarrow A $, 
that is, a multiplication in the coefficient group  $  A $(
see [[#References|[2]]]). The  $  \nu _ {m,n} $
induce mappings

$$ 
 \mathop{\rm Hom} ( X _ {m} , A ) \otimes  \mathop{\rm Hom} ( X _ {n} , A )
 \rightarrow   \mathop{\rm Hom} ( X _ {m+n} , A ) ,
$$

which in their turn induce mappings  $  \nu _ {m,n} $
in cohomology.

In particular, a ring structure is defined on the graded group  $  H ( G , \mathbf Z ) = \oplus_{n=0}^  \infty  H  ^ {n} ( G , \mathbf Z ) $, 
where  $  G $
is a group and  $  \mathbf Z $
is the ring of integers with a trivial  $  G $-
action. The corresponding mappings  $  \nu _ {m,n} $
coincide with the  $  \cup $-
product. This is an associative ring with identity, and for homogeneous elements  $  a , b \in H ( G , \mathbf Z ) $
of degrees  $  p , q $
respectively,  $  a b = ( - 1 )  ^ {pq} b a $.

Analogously, the  $  \cup $-
product defines a ring structure on the group  $  \oplus_{n=0}^  \infty  H  ^ {n} ( X , \mathbf Z ) $, 
where  $  H  ^ {n} ( X , \mathbf Z ) $
is the  $  n $-
dimensional singular cohomology group of a topological space  $  X $
with coefficients in  $  \mathbf Z $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cartan,   S. Eilenberg,   "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. MacLane,   "Homology" , Springer  (1963)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Dold,   "Lectures on algebraic topology" , Springer  (1972)  pp. Chapt. VII</TD></TR></table>