# 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.

Expert's answer

First assume that some isotypiccomponent, say

Next, assume all

We conclude that

(product)

*M*1, is not finitely generated. Then*M*1 is an*infinite*direct sum of a simple*k*-module, so it is easy to find anepimorphism*f*1 :*M*1*→ M*1 which is not an isomorphism.Extending*f*1 by the identity map on the other*Mi*’s, we get an*f*:*M → 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*’s arefinitely generated.*M*=*K**⊕**N*=*K'**⊕**N'*and*N**∼**N'**⇒**K**∼**K'.*We conclude that

*R*isunit-regular. Alternatively, we can give a more direct argument. Since*Mi*isfinitely generated,*Ri*: = End*k*(*Mi*) is a simple artinianring, so it is unit-regular. It is easy to see that*R*= End*k*(*⊕**iMi*)*∼*(product)

*Ri,*so it followsthat*R*is also unit-regular.
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