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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 $\displaystyle\bigoplus_{i=1}^{q} A_{i}$?

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up vote 3 down vote accepted

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 .

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Dear @user, the reference is: Camps and Dicks, Israel Journal of Mathematics Volume 81, Numbers 1-2, 203-211. And, no, "just taking quotients" won't work in general: check the concept "stably isomorphic but not isomorphic". –  Georges Elencwajg Apr 2 '12 at 11:17
    
The flaw is that you may not quotient common summands in isomorphisms: else you would have cancellation for all modules over all rings! –  Georges Elencwajg Apr 2 '12 at 18:10
    
Dear user, the (admittedly subtle) point is that if $M=A_1\oplus A_2$ (i.e. $A_i\subset M $) , you may deduce that $M/A_1\cong A_2$ but you may not if you only have $M\cong A_1\oplus A_2$. Think it over! If you still have difficulties, ask a new question: this thread is already very long... –  Georges Elencwajg Apr 3 '12 at 23:23

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