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Some remarks about elementary divisor rings

1956; American Mathematical Society; Volume: 82; Issue: 2 Linguagem: Inglês

10.1090/s0002-9947-1956-0078979-8

1088-6850

Leonard Gillman, Melvin Henriksen,

Algebraic structures and combinatorial models

In this and the following paper [2], we are concerned with obtaining conditions on a commutative ring S with identity element in order that every matrix over S can be reduced to an equivalent diagonal matrix(2).Following Kaplansky [4], we call such rings elementary divisor rings.A necessary condition is that 5 satisfy F: all finitely generated ideals are principal.It has been known for some time that if 5 satisfies the ascending chain condition on ideals, and has no zero-divisors, then F is also sufficient.Helmer [3] showed that the chain condition can be replaced by the less restrictive hypothesis that S be adequate (i.e., of any two elements, one has a "largest" divisor that is relatively prime to the other(2)).Kaplansky[4] generalized this further by permitting zero-divisors, provided that they are all in the (Perlis-Jacobson) radical.By a slight modification of Kaplansky's argument, we find that the condition on zero-divisors can be replaced by the hypothesis that 5 be an Hermite ring (i.e., every matrix over 5 can be reduced to triangular form(2)).This is an improvement, since, in any case, it is necessary that 5 be an Hermite ring, while, on the other hand, it is not necessary that all zerodivisors be in the radical.In fact, we show that every regular commutative ring with identity is adequate.However, the condition that 5 be adequate is not necessary either.We succeed in obtaining a necessary and sufficient condition that S be an elementary divisor ring.Along the way, we obtain a necessary and sufficient condition that S be an Hermite ring.In the paper that follows [2], we make constant use of these results.In particular,we construct examples of rings that satisfy F but are not Hermite rings, and examples of Hermite rings that are not elementary divisor rings.However, all these examples contain zero-divisors; therefore, the question as to whether there exist corresponding examples that are integral domains is left unsettled.Definition 1.An m by re matrix A over 5 admits triangular reduction ifPresented to the Society, December 28, 1954 under the title Concerning adequate rings and elementary divisor rings.II;