Cancelling summands in a direct sum decomposition Let $M$ be a Noetherian and Artinian module. Suppose that:
$$\bigoplus_{i=1}^{q} A_{i} \oplus \bigoplus_{i=1}^{t} B_{i} \cong  \bigoplus_{i=1}^{q} A_{i} \oplus \bigoplus_{i=1}^{r} C_{i}$$
where all $A_{i},B_{i},C_{i}$ are indecomposable submodules of $M$.
Can we always guarantee that $B_{i} \cong C_{i}$ for all $i \in \{1,2,...,t\}$? That is, can we "cancel" the term $\displaystyle\bigoplus_{i=1}^{q} A_{i}$?

Cancellation means that for modules   $M,N,P$ over a ring $R$ (not assumed commutative) we have the implication 
$$M\oplus N\cong M\oplus P \implies N\cong P$$
Cancellation holds for modules that are only  assumed artinian  (which of course answers your question in the affirmative) thanks to a theorem by Camps and Dicks.
This is quite astonishing, since 
 Krull-Schmidt does not hold for modules that are just supposed artinian.
And, again  astonishingly, a counter-example was found only in 1995 .    
Finally, let me point out that a very general Krull-Schmidt theorem was proved in a categorical setting by Atiyah. The main application of his results is to coherent sheaves in  algebraic geometry .
