By Jean-Paul Pier
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Deduce that I + AB is a unit if and only if I + BA is; prove this directly by evaluating I − B(I + AB)−1 A. 4◦ . Under what circumstances is AB stably associated to BA? 5. Let R be a ring with UGN. If A, A satisfy a comaximal relation (7), show that i(A) ≥ i(A ). Deduce that if A, A satisfy a comaximal relation and A , A likewise, then A and A are stably associated. 6. 7. Hence find examples of pairs of matrices (over a weakly finite ring, say) that satisfy a comaximal relation but are not stably associated.
For any ring R, show that R and Rn (n > 1) have isomorphic centres. Prove this fact by characterizing the centre of R as the set of all natural transformations of the identity functor on R Mod. 7. Let R = K n be a full matrix ring and f : R → S any ring homomorphism. Show that S is a full n × n matrix ring, say S = L n and there is a homomorphism φ : K → L inducing f. 8. (G. M. Bergman) Let n ≥ 1 be an integer and R a ring in which every right ideal that is not finitely generated has a finitely generated direct summand that cannot be generated by n elements.
3 Projective modules 15 The structure of S(R) is closely related to certain properties of the ring R, while K o (R) reflects the corresponding stable properties. This is illustrated in the next result, where by a stably free module we understand a module P such that P ⊕ R m ∼ = R n , for some integers m, n ≥ 0. 4. Let R be any ring and denote by λ : Z → K o (R) the homomorphism mapping 1 to (R). Then (i) R has IBN if and only if λ is injective and (ii) every finitely generated projective module is stably free if and only if λ is surjective.