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Representation of an irreducible direct sum of simple modules as tensor product

Let $k$ be a field and $A$ a finite-dimensional algebra over $k$.
Fix an infinite set of simple modules $\{M_i\}_{i\in I}$ for $A$ (one can think of choosing a basis).
There is a well-defined tensor product $\otimes$ on the category of finitely generated left $A$-modules.
Moreover, we have a canonical way to represent each simple module $M$ as an $A$-submodule of $\otimes M$.
Question: Does there exists a basis of $M$ over $k$ such that $M$ is in «canonical form» in the sense that every nonzero simple $A$-module appears as a simple submodule in $\otimes M$ with nonzero multiplicity (equivalently, as a direct summand in the sum of $A$-modules $\oplus M_i$)?


Yes! Write $M$ as a tensor product of a finite number of simple submodules $M_i$, and assume $M_i
e 0$. Then it is relatively easy to see that $M_i$ is isomorphic to a direct sum of copies of $M$.
This is the result of a theorem of Peter Gabriel (


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The results of the treatment of 132 patients with cancer of the stomach were reviewed and its influence on the 5-year survival of these patients was evaluated. There was a statistically significant increase of the 5-year survival in the cases with early stages of the disease, absence of specific symptoms and recovery of ECOG performance status after the therapy. The best results were achieved in the patients who had the cancer resected. The possibility of a radical resection depended on the stage of the disease and on the application of preoperative therapy.Electrophysiological measurement of direction of response in the auditory cortex of alert monkeys.
A neuron or a columnar cluster of neurons is essential to process the direction of a sound. We have recorded the sound-evoked multielectrode array responses from the auditory cortex of alert monkeys with an acoustical algorithm. We found 14 ipsilateral and 23 contralateral neuron clusters and 16 pairs of simultaneously recorded neurons. Fourteen of the neuron clusters showed responses exclusively for air-conducted stimuli, and only one cluster showed responses for bone-conducted stimuli. The mean receptive field (RF) size of the neuron clusters was 1.82 +/- 1.20 degrees (mean +/- SD) for air-conducted stimuli and 3.70 +/- 4.05 degrees for bone-conducted stimuli. The RF size of neuron pairs that differed significantly between each other was smaller than those in a neuron cluster, being 0.47 +/- 0.28 degrees for the air-conducted stimuli and 1.09 +/- 0.85 degrees for the bone-conducted stimuli. The RF size decreased with the distance from the wall of the recording chamber. The mean refractory period was 38.3 +/- 13.2 ms for the air-conducted and 54.0 +/- 8.3 ms for the bone-conducted stimuli. The low percentage of neurons that were less sensitive to sound with bone-conducted stimuli (4.7%) compared with air