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Let A be a finite-dimensional algebra over an algebraically closed field \(\Bbbk \). For any finite-dimensional A-module M we give a general formula that computes the indecomposable decomposition of M 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 Mróz. As an example we apply this formula to the Kronecker algebra A and give an explicit formula to compute the indecomposable decomposition of M, which enables us to make a computer program.

Decomposition theory of modules: the case of Kronecker algebra