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# Answer to Question #17368 in Algebra for Melvin Henriksen

Question #17368
Let M be a semisimple right k-module. Show that R = Endk(M) is unit-regular iff the isotypic components Mi of M are all finitely generated.
First assume that some isotypiccomponent, say M1, is not finitely generated. Then M1 is an infinitedirect sum of a simple k-module, so it is easy to find anepimorphism f1 : M1 &rarr; M1 which is not an isomorphism.Extending f1 by the identity map on the other Mi&rsquo;s, we get an f: M &rarr; M which is an epimorphism but not an isomorphism. Now for anysplitting g for f, we have fg = 1 not equal gf.So, R cannot be unit-regular.
Next, assume all Mi&rsquo;s arefinitely generated. M = K &oplus; N = K&#039; &oplus; N&#039; and N &sim; N&#039; &rArr; K &sim; K&#039;.
We conclude that R isunit-regular. Alternatively, we can give a more direct argument. Since Mi isfinitely generated, Ri : = Endk(Mi) is a simple artinianring, so it is unit-regular. It is easy to see that R = Endk(&oplus;iMi) &sim;
(product)Ri, so it followsthat R is also unit-regular.

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