By Atkins P.W., Child M.S., Phillips S.G.

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Obtain the same conclusion for DCCn . 2 Matrix rings and the matrix reduction functor 7 6. Let R be a non-zero ring without IBN and for fixed m, n(m = n) consider pairs of mutually inverse matrices A ∈ m R n , B ∈ n R m . Show that if A , B is another such pair, then P = A B is an invertible matrix such that P A = A , BP −1 = B . What is P −1 ? 7. Let R be a weakly n-finite ring. Given maps α : R r → R n and β : R n → R s (r + s = n) such that αβ = 0, α has a right inverse and β has a left inverse, then there exists an automorphism μ of R n such that αμ : R r → R n is the natural inclusion and μπ = β, where π : R n → R s is the natural projection.

Show that conversely, every ring with this property is weakly n-finite. ) 8. Show that a ring R is weakly n-finite if and only if (F): Every surjective endomorphism of R n is an automorphism. If a non-zero ring R has the property (F), show that every free homomorphic image of R n has rank at most n. Deduce that every non-zero weakly finite ring has UGN. 9∗ . Which of IBN, UGN, weak finiteness (if any) are Morita invariants? 10◦ . Characterize the rings all of whose homomorphic images are weakly finite.

11. Let R be a ring. If the product of any two full matrices of the same size is full and any full matrix is stably full, then R is projectivefree. 3 1. Verify the equivalence of the two definitions of S(R), in terms of projective modules and idempotent matrices. 4 Hermite rings 19 2. Let R be a ring and J = J (R). By considering the kernel of the homomorphism Rn → (R/J )n induced by the natural homomorphism R → R/J , show that J (Rn ) ∼ = Jn . 3◦ . Show that if R is a matrix local ring and n > 1, then so is Rn .