Elementary divisor domain
This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
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VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions
History
Origin
This term was introduced by: Kaplansky
The notion of an elementary divisor domain was introduced by Kaplansky in his paper Elementary Divisors and Modules.
It was later generalized to elementary divisor ring by Gillman and Henriksen, in Some remarks about elementary divisor rings.
Definition
Symbol-free definition
An elementary divisor domain or Hermite domain is any of the following equivalent things:
- An integral domain which is also an elementary divisor ring
- An integral domain which is also a Hermite ring
Definition with symbols
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Relation with other properties
Stronger properties
References
- Elementary divisors and modules by I. Kaplansky, Trans. Amer. Math. Society, 66 (1949), 464-491
- Some remarks about elementary divisor rings by L. Gillman and M. Henriksen, Trans. Amer. Math. Society, 82 (1956), 362-365