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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’.
For a &isin; R, write M = ker(a) &oplus; P = Q &oplus; im(a).Since a defines an isomorphism from P to im(a), thehypothesis implies that ker(a) &sim; Q(&sim; coker(a)). Defining u &isin; U(R) such that u is an isomorphism from Q to ker(a),and u : im(a) &rarr; P is the inverse of a|P : P &rarr; im(a),we have a = aua &isin; R. For thenecessity part, assume R is unit-regular. Suppose M = K &oplus; N = K&#039; &oplus; N&#039;, where N &sim; N&#039;. Define a &isin; R such thata(K) = 0 and a|N is a fixed isomorphism from N to N&#039;.Write a = aua, where u &isin; U(R).
(&lowast;) M = ker(a) &oplus; im(ua) = K &oplus; u(N&#039;).
Since u defines anisomorphism from N&#039; to u(N&#039;), it induces an isomorphismfrom M/N&#039; to M/u(N&#039;). Noting that M/N&#039; &sim; K&#039; and M/u(N&#039;) &sim;K (from (&lowast;)), weconclude that K &sim; K&#039;.

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