By Murakami M.

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Show that any Artinian matrix local ring is a full matrix ring over a scalar local ring. g. ) 9. Let R be the ring of rational quaternions with denominator prime to p, an odd prime. Show that the Jacobson radical of R is p R and R/ p R is the ring of quaternions over F p . Deduce that R is a matrix local ring which is not a matrix ring over a scalar local ring. 10. Show that for any ring R the following are equivalent (see Lorimer [92]): (a) R is local and any finitely generated left ideal is principal, (b) the principal left ideals of R are totally ordered by inclusion, (c) all left ideals of R are totally ordered by inclusion.

Prove the converse when R is Hermite. 3. –6. are Morita invariant? For the others describe the rings that are Morita invariant to them. 4. If in an Hermite ring, AB = I and B is completed to an invertible matrix (B, B ), show that for suitably chosen A , (A, A )T has the inverse (B, B − B AB ). 5. Given A ∈ mR n , B ∈ nR m , where m < n, over any ring R, such that AB = Im , show that A is completable if and only if A:0 = {x ∈ nR|Ax = 0} is free of rank n − m (Kazimirskii and Lunik [72]). 6. Define an n-projective-free ring as a ring over which every n-generator projective module is free of unique rank.

Writing Q = M/N , we have a natural ring homomorphism I (N ) → EndR (Q); the kernel is easily seen to be a, so we obtain an injection E(N ) → End R (Q). (1) Suppose now that M is projective. Then any endomorphism φ of Q can be lifted to an endomorphism β of M such that Nβ ⊆ N ; this shows the map (1) to be surjective, and so an isomorphism. 1. Given any ring R, if P is a projective left R-module and N a submodule of P with eigenring E(N), then there is a natural isomorphism E(N ) ∼ = End R (P/N ).