## Document Type

Article

## Department

Mathematics (HMC)

## Publication Date

1956

## Abstract

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 [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.

## Rights Information

© 1956 American Mathematical Society

## Terms of Use & License Information

## DOI

10.1090/S0002-9947-1956-0078979-8

## Recommended Citation

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

## Comments

Previously linked to as: http://ccdl.libraries.claremont.edu/u?/irw,406.

Pdf created from original.

This article is also available at http://www.ams.org/journals/tran/1956-082-02/S0002-9947-1956-0078979-8/.