By a slight modification of Kaplansky's argument, we find that the condition on zero-divisors can be replaced by the hypothesis that S be an Hermite ring (i.e., every matrix over S can be reduced to triangular form). This is an improvement, since, in any case, it is necessary that S be an Hermite ring, while, on the other hand, it is not necessary that all zero-divisors be in the radical. In fact, we show that every regular commutative ring with identity is adequate. However, the condition that S 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 , 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.
© 1956 American Mathematical Society
Gillman, L., and M. Henriksen. "Some remarks about elementary divisor rings." Transactions of the American Mathematical Society 82 (1956): 362-365. DOI: 10.1090/S0002-9947-1956-0078979-8