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On Divisibility and Injectivity

Published online by Cambridge University Press:  20 November 2018

Garry Helzer
Garry Helzer*
Affiliation:
University of Maryland, College Park, Md
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In the category of abelian groups the concepts of divisible group and injective group coincide. In (4) this is generalized to modules over an integral domain and it is proved for a (commutative) integral domain that the concepts of divisible module and injective module coincide if and only if the ring is hereditary if and only if the ring is a Dedekind domain.

In (8) the assumption of commutativity is dropped and the ring is assumed to have a one-sided field of quotients. The result obtained is essentially the same as in the commutative case; see the theorem following 6.2. In (13) the requirement that the ring have no zero-divisors is also dropped and the ring is assumed to possess what we have called an Ore ring (see the definition following 6.2). The result obtained is the equivalent of parts (a) and (b) of our 6.13.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 901 - 919
Copyright
Copyright © Canadian Mathematical Society 1966

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