A divisible module is a module over a unit ring such that for all which are not zero divisors, every element of can be "divided" by , in the sense that there is an element in such that . This condition can be reformulated by saying that the multiplication by defines a surjective map from to .

It can be shown that every injective -module is divisible, but the converse only holds for particular classes of rings, e.g., for principal ideal domains. Since and are evidently divisible -modules, this allows us to conclude that they are also injective.

An additive Abelian group is called divisible if it is so as a -module.

See also

Injective Module

This entry contributed by Margherita Barile

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References

Beachy, J. A. Introductory Lectures on Rings and Modules. Cambridge, England: Cambridge University Press, p. 97, 1999.Bruns, W. and Herzog, J. Cohen-Macaulay Rings, 2nd ed. Cambridge, England: Cambridge University Press, p. 90, 1998.Faith, C. Algebra: Rings, Modules and Categories, I. Berlin, Germany, pp. 158-159, 1973.Hilton, P. J. and Stammbach, U. A Course in Homological Algebra, 2nd ed. New York: Springer-Verlag, pp. 31-33, 1997.Rowen, L. H. Ring Theory, Vol. 1. San Diego, CA: Academic Press, pp. 263-266, 1988.

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Divisible Module

Cite this as:

Weisstein, Eric W., with contributions by Margherita Barile. "Divisible Module." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DivisibleModule.html

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