TITLE: What is a "block" in an abelian category?
QUESTION [11 upvotes]: In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which are categories of modules by Freyd-Mitchell).     This raises the natural question:

What is meant by a "block" in an abelian category?

The concept originates to some extent in the modular representation theory of finite groups or their group algebras pioneered by Richard Brauer.
Here a block is just an indecomposable two-sided ideal of the group algebra,
corresponding to a primitive central idempotent.   But in later developments the
language of homological algebra plays a greater role than the group algebra or its center: the category of modules decomposes into a direct sum of subcategories,
which are as small as possible relative to permitting no nontrivial extensions among their simple objects (irreducible representations).    This approach  generalizes well to other situations, where a center or central characters may be elusive and where the decomposition may be infinite, etc.   By now "blocks" occur in many areas of representation theory influenced by classical Lie theory: algebraic groups, restricted enveloping algebras, quantum analogues,
finite $W$-algebras, Cherednik algebras, Kac-Moody algebras and groups, Lie superalgebras.
There is some inconsistency in the literature about allowing "blocks" which might be further decomposed into direct sums.    In classical or Kac-Moody Lie theory
this usually reflects the special influence of infinitesimal/central characters.    But full centers in Lie theory and related quantum groups may be unknown or unneeded, e.g., the approach Kac took to his analogue of the Weyl character formula for integrable modules of an affine Lie algebra relied just on a single Casimir-type operator (which works equally well in the classical finite-dimensional case).  In Jantzen's book Representations of Algebraic Groups (AMS, 2003), the discussion of blocks  for algebraic group schemes in II.7.1 is careful but not completely general.
In practice looser definitions of "block" than the homological one work well enough in many settings, but it creates some confusion when the word is used with no definition at all.  Is there a single convention which reduces in familiar cases to older usage?   In a recent comment to another post I paraphrased  Humpty Dumpty ("my remote ancestor"), who actually said: "When I use a word it means just what I choose it to mean --- neither more nor less." 
But communication is better when the short and convenient word "block" starts out with a common meaning.

REPLY [7 votes]: The classic situation in which blocks are defined is for an artinian ring $A$.  In this case, writing $1=e_1+\cdots+e_n$ as a sum of central primitive idempotents, the blocks are the (two-sided) ideals $A_i=Ae_i$.  For each non-zero indecomposable module $M$, there is a unique $1 \leq i \leq n$ for which $e_iM \neq 0$, and in this case $M$ belongs to the block $A_i=eA_i$.  In particular this partitions the set of irreducible $A$-modules into blocks (more categorically, decomposes $A$-mod as a direct sum of $A_i$-mod's).
There are (at least) two ways in which one might try to generalize this setup:
(1) Weaken the Artinian hypothesis.
(2) Weaken the hypothesis that we are looking at the category of modules for a ring. 
It looks to me like (2) is treated in Torsten's answer, but it looks like no one has mentioned the (by now standard, at least for Noetherian ring theorists) answer for (1): here the replacement for blocks is "cliques". 
Suppose $R$ is a Noetherian ring.  "Ideal" will mean two-sided unless otherwise specified.  Prime ideals $P$ and $Q$ of $R$ are linked if there is a an ideal $I$ with $QP \subseteq I \subset P \cap Q$, and $P \cap Q/I$ non-zero, torsion free as a right $R/P$-module and as a left $R/Q$-module.  The cliques of $R$ are the connected components of the graph with vertex set $\mathrm{Spec}(R)$ and with edges given by links.  (The graph is really directed b/c of the assymetry in the definition: really it should be an arrow from $Q$ to $P$ but for defining cliques this makes no difference). 
To make the definition a little more intuitive, assume $R$ is an algebra over a field $F$, that $M$ and $N$ are irreducible $R$-modules which are $F$-finite dimensional, and that $P$ and $Q$ are the annihilators of $M$ and $N$.  Then it should be a not-too-hard exercise to check that $P$ and $Q$ are linked iff there is a non-trivial extension between $M$ and $N$.
In an Artinian ring, the cliques are the same thing as the blocks (see e.g. pages 142 to 144 of Jategaonkar's book "Localization in Noetherian Rings").  For Noetherian ring theory, the original purpose of introducing cliques was to figure out which multiplicative subsets were "good" for localizing (K. Brown's survey "Ore Sets in Noetherian Rings" is readable and discusses this motivation).