By Tanizaki H.

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Show that a ring with left ACCn for some n ≥ 1 is weakly n-finite. Deduce that a left (or right) Noetherian ring, or more generally, a ring with left (or right) panACC is weakly finite. ) 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 .

Beck [72]) Let P be a finitely generated projective left R-module. If P/JP is free over R/J , where J is the Jacobson radical of R, show that P is free over R. ) 12∗ . (Kaplansky [58]) Let P be a projective module over a local ring. Show that any element of P can be embedded in a free direct summand of P; deduce that every projective module over a local ring is free. 1 hold in most rings normally encountered, and counter-examples belong to the pathology of the subject. By contrast, the property defined below forms a significant restriction on the ring.

Hence m = 0, Q = 0 and n P∼ =R . 10. Let R be a ring. If for every n ≥ 1 the product of two full n × n matrices is again full, then every finitely generated projective module is stably free. Proof. Suppose there is a finitely generated projective module that is not stably free; we choose such a module P with the least number of generators, n say. 3 and the minimality of n. The module Q = R n (I − E) is such that P ⊕ Q = R n , and since E(I − E) = 0, I − E cannot be full, so Q can be generated by fewer than n elements and hence is stably free.

### A Simple Gamma Random Number Generator for Arbitrary Shape Parameters by Tanizaki H.

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