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A functor  "measuring"  the deviation of a given functor from being exact. Let  $  T ( A , C ) $
be an additive functor from the product of the category of  $  R _ {1} $-modules with the category of  $  R _ {2} $-modules into the category of  $  R $-modules that is covariant in the first argument and contravariant in the second argument. From an injective resolution  $  X $
of  $  A $
and a projective resolution  $  Y $
of  $  C $
one obtains a doubly-graded complex  $  T( X , Y ) $. 
The homology of the associated single complex  $  T ( A , C ) $
does not depend on the choice of resolutions, has functorial properties and is called the right derived functor  $  R  ^ {n} T ( A , C ) $
of  $  T ( A , C ) $. 
The basic property of a derived functor is the existence of long exact sequences

$$ 
\rightarrow  R  ^ {n} T ( A  ^  \prime  , C )  \rightarrow  R  ^ {n} T ( A , C )  \rightarrow  R  ^ {n}
T ( A  ^ {\prime\prime} , C ) \rightarrow
$$

$$ 
\rightarrow \ 
R  ^ {n+ 1} T ( A  ^  \prime  , C )  \rightarrow \dots
$$

$$ 
\rightarrow  R  ^ {n} T ( A , C  ^ {\prime\prime} )  \rightarrow  R  ^ {n} T (
A , C )  \rightarrow  R  ^ {n} T ( A , C  ^  \prime  ) \rightarrow
$$

$$ 
\rightarrow \ 
R  ^ {n+ 1} T ( A , C  ^ {\prime\prime} )  \rightarrow \dots ,
$$

induced by short exact sequences

$$ 
0  \rightarrow  A  ^  \prime   \rightarrow  A  \rightarrow  A  ^ {\prime\prime}  \rightarrow  0,
$$

$$ 
0  \rightarrow  C  ^  \prime   \rightarrow  C  \rightarrow  C  ^ {\prime\prime}  \rightarrow  0 .
$$

The left derived functor is defined analogously. The derived functor of  $   \mathop{\rm Hom} _ {R} $
is denoted by  $   \mathop{\rm Ext} _ {R}  ^ {n} $. 
The group  $   \mathop{\rm Ext} _ {R}  ^ {1} ( A , C ) $
classifies extensions of  $  A $
with kernel  $  C $
up to equivalence (cf. [[Baer multiplication|Baer multiplication]]; [[Cohomology of algebras|Cohomology of algebras]]).

====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></table>

====Comments====
The above article does not explain the sense in which  $  R  ^ {n} T $
measures the deviation of  $  T $
from being exact. The point is that if  $  T $
is left exact (i.e. preserves the exactness of sequences of the form  $  0 \rightarrow A  ^  \prime  \rightarrow A \rightarrow A  ^ {\prime\prime} $
in the fist variable and of the form  $  C  ^  \prime  \rightarrow C \rightarrow C  ^ {\prime\prime} \rightarrow 0 $
in the second), then  $  R  ^ {0} T $
is naturally isomorphic to  $  T $; 
if further  $  T $
is exact, then  $  R  ^ {n} T = 0 $
for all  $  n > 0 $. 
Derived functors may also be defined for additive functors of a single variable between module categories, or, more generally, between arbitrary Abelian categories, provided the necessary injective or projective resolutions exist in the domain category.