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An [[Ideal|ideal]]  $  I $
of a ring  $  A $
is said to be indecomposable if, for any ideals  $  X $
and  $  Y $
of  $  A $, 
$  I = X \oplus Y $
implies  $  X = 0 $
or  $  Y = 0 $. 
The ideal  $  I $
is called a direct summand of  $  A $
if  $  A = I \oplus J $
for some ideal  $  J $
of  $  A $. 
A block of  $  A $
is defined to be any ideal of  $  A $
which is an indecomposable direct summand of  $  A $. 
By a block idempotent of  $  A $
one understands any primitive [[Idempotent|idempotent]] of the centre of  $  A $(
cf. also [[Centre of a ring|Centre of a ring]]). An ideal  $  B $
of  $  A $
is a block of  $  A $
if and only if  $  B = Ae $
for some (necessarily unique) block idempotent  $  e $
of  $  A $. 
Thus blocks and block idempotents determine each other.

Any decomposition of  $  A $
of the form  $  A = B _ {1} \oplus \dots \oplus B _ {n} $, 
where each  $  B _ {i} $
is a block of  $  A $, 
is called a block decomposition of  $  A $. 
In general, such a decomposition need not exist, but it does exist if  $  A $
is semi-perfect (cf. [[Perfect ring|Perfect ring]]). In the classical case where  $  A $
is semi-primitive Artinian (cf. [[Primitive ring|Primitive ring]]; [[Artinian ring|Artinian ring]]), each block of  $  A $
is a complete matrix ring over a suitable division ring, and the number of blocks of  $  A $
is equal to the number of non-isomorphic simple  $  A $-
modules.

The study of blocks is especially important in the context of group representation theory (see [[Representation of a group|Representation of a group]]; [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]]). Here, the role of  $  A $
is played by the group algebra  $  RG $, 
where  $  G $
is a [[Finite group|finite group]] and the commutative ring  $  R $
is assumed to be a complete Noetherian semi-local ring (cf. also [[Commutative ring|Commutative ring]]; [[Noetherian ring|Noetherian ring]]; [[Local ring|Local ring]]) such that  $  R/J ( R ) $
has prime characteristic  $  p $. 
The most important special cases are:

$  R $
is a complete discrete valuation ring of characteristic  $  0 $
with  $  R/J ( R ) $
of prime characteristic  $  p $;

$  R $
is a field of prime characteristic  $  p $.

One of the most useful aspects of modular representation theory is the study of the distribution of the irreducible ordinary characters of  $  G $
into blocks. The main idea is due to R. Brauer and can be described as follows. Let  $  G $
be a [[Finite group|finite group]] and let  $  p $
be a prime number. Assume that  $  R $
is a complete discrete [[Valuation|valuation]] ring of characteristic  $  0 $, 
$  K $
is the quotient field of  $  R $
and  $  R/J ( R ) $
is of characteristic  $  p $. 
Let  $  { \mathop{\rm Irr} } ( G ) $
be the set of all irreducible  $  K $-
characters of  $  G $(
cf. [[Character of a group|Character of a group]]) and write  $  B = B ( e ) $
to indicate that  $  B $
is a block of  $  RG $
whose corresponding block idempotent is  $  e $, 
i.e.,  $  B = RGe $. 
The character  $  \chi \in { \mathop{\rm Irr} } ( G ) $
is said to belong to the block  $  B = B ( e ) $
of  $  RG $
if  $  \chi ( e ) \neq 0 $(
here  $  \chi $
is extended by  $  K $-
linearity to the mapping  $  \chi : {KG } \rightarrow K $). 
It turns out that if  $  B _ {1} \dots B _ {n} $
are all distinct blocks of  $  RG $, 
then  $  { \mathop{\rm Irr} } ( G ) $
is a disjoint union of the  $  { \mathop{\rm Irr} } ( B _ {i} ) $, 
$  1 \leq  i \leq  n $, 
where  $  { \mathop{\rm Irr} } ( B _ {i} ) $
denotes the set of irreducible  $  K $-
characters of  $  G $
belonging to  $  B _ {i} $. 
In the classical case studied by Brauer, namely when  $  K $
is a splitting field for  $  G $, 
the irreducible  $  K $-
characters of  $  G $
are identifiable with the irreducible  $  \mathbf C $-
characters of  $  G $.

Assume that, in the context of the previous paragraph,  $  K $
is a splitting field for  $  G $. 
Let  $  B $
be a block of  $  RG $
and let  $  p  ^ {a} $
be the order of Sylow  $  p $-
subgroups of  $  G $(
cf. [[Sylow subgroup|Sylow subgroup]]). It turns out that there exists an integer  $  d \geq  0 $, 
called the defect of  $  B $, 
such that  $  p ^ {a - d } $
is the largest power of  $  p $
which divides  $  \chi ( 1 ) $
for all  $  \chi \in { \mathop{\rm Irr} } ( B ) $. 
The notion of the defect of  $  B $
can be defined by purely ring-theoretic properties under much more general circumstances. Namely, it suffices to assume that  $  R $
is a complete Noetherian semi-local ring such that  $  R/J ( R ) $
has prime characteristic  $  p $(
see [[#References|[a5]]]; [[Noetherian ring|Noetherian ring]]).

For the classical case where  $  K $
is a splitting field for  $  G $, 
one has the following famous problem, frequently called the Brauer  $  k ( B ) $-
conjecture. Let  $  B $
be a block of  $  RG $
and let  $  k ( B ) = | { { \mathop{\rm Irr} } ( B ) } | $. 
Is it true that  $  k ( B ) \leq  p  ^ {d} $, 
where  $  d $
is the defect of  $  B $? 
Although many special cases have been attacked successfully, the general case is still far from being solved. So far (1996), Brauer's  $  k ( B ) $-
conjecture has not been verified for all finite simple groups. Moreover, it is not known whether Brauer's  $  k ( B ) $-
conjecture can be reduced to the case of simple (or at least quasi-simple) groups (see [[#References|[a5]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Karpilovsky,   "Group representations" , '''1''' , North-Holland  (1992)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Karpilovsky,   "Group representations" , '''2''' , North-Holland  (1993)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Karpilovsky,   "Group representations" , '''3''' , North-Holland  (1994)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Karpilovsky,   "Group representations" , '''4''' , North-Holland  (1995)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G. Karpilovsky,   "Group representations" , '''5''' , North-Holland  (1996)</TD></TR></table>