# Answer to Question #17367 in Algebra for Melvin Henriksen

Question #17367

Let M be a right module over a ring k such that R = Endk(M) is von Neumann regular. Show that R is unit-regular iff, whenever M = K ⊕ N = K’ ⊕ N’ (in the category of k-modules), N ∼ N’ implies K ∼ K’.

Expert's answer

For

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Since

*a**∈**R*, write*M*= ker(*a*)*⊕**P*=*Q**⊕**im(**a*).Since*a*defines an isomorphism from*P*to im(*a*), thehypothesis implies that ker(*a*)*∼**Q*(*∼*coker(*a*)). Defining*u**∈**U(**R*) such that*u*is an isomorphism from*Q*to ker(*a*),and*u*: im(*a*)*→ P*is the inverse of*a|P*:*P →*im(*a*),we have*a*=*aua**∈**R*. For thenecessity part, assume*R*is unit-regular. Suppose*M*=*K**⊕**N*=*K'**⊕**N',*where*N**∼**N'*. Define*a**∈**R*such that*a*(*K*) = 0 and*a|N*is a fixed isomorphism from*N*to*N'*.Write*a*=*aua*, where*u**∈**U(**R*).(

*∗*)*M*= ker(*a*)*⊕**im(**ua*) =*K**⊕**u*(*N'*)*.*Since

*u*defines anisomorphism from*N'*to*u*(*N'*), it induces an isomorphismfrom*M/N'*to*M/u*(*N'*). Noting that*M/N'**∼**K'*and*M/u*(*N'*)*∼**K*(from (*∗*)), weconclude that*K**∼**K'*.
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