Walker groups

نویسنده

Fred Richman

چکیده

A reformulation of Walker’s theorem on the cancellation of Z says that any two homomorphisms from an abelian group B onto Z have isomorphic kernels. It does not have a constructive proof, even for B a subgroup of Z3. In this paper we give a constructive proof of Walker’s theorem for B a direct sum, over any discrete index set, of groups of the following two kinds: finite-rank torsion-free groups whose finite-rank torsion-free quotients have computable heights, and Butler groups with weakly computable heights. Throughout, “group”means abelian group. The infinite cyclic group, and the ring of integers, is denoted by Z. The nonnegative integers are denoted by N, the positive integers by Z, and the rational numbers by Q. We say that a set is discrete if any two elements are either equal or different. A subset A of a set B is detachable (from B) if for each b ∈ B, either b ∈ A or b / ∈ A. A group is discrete if and only if its subset {0} is detachable. Elbert Walker, in his dissertation [7], and P. M. Cohn in [2], showed that Z is cancellable in the sense that if Z ⊕ B ∼= Z ⊕ B′, then B ∼= B′. It is somewhat of an oddity that Z is cancellable. A rank-one torsion-free group A is cancellable if and only if A ∼= Z or the endomorphism ring of A has stable range one [3], [1, Theorem 8.12]. (A ring R has stable range one if whenever aR + bR = R, then a + bR contains a unit of R.) In fact, any object in an abelian category whose endomorphism ring has stable range one is cancellable. The endomorphism ring of Z does not have stable range one, so Z is the unique rank-one torsion-free group that is cancellable for some reason other than its endomorphism ring.

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