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    Decomposition theory of modules: the case of Kronecker algebra

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    Let AA be a finite-dimensional algebra over an algebraically closed field k\Bbbk. For any finite-dimensional AA-module MM we give a general formula that computes the indecomposable decomposition of MM without decomposing it, for which we use the knowledge of AR-quivers that are already computed in many cases. The proof of the formula here is much simpler than that in a prior literature by Dowbor and Mr\'oz. As an example we apply this formula to the Kronecker algebra AA and give an explicit formula to compute the indecomposable decomposition of MM, which enables us to make a computer program
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