Integral domain
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Let R be an integral domain. An R-module M is called divisible if for every x in M and every nonzero r in R, there exists y in M such that ry=x...(see attached)
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Solution Summary
This provides an example of proving divisibility with R-modules in integral domains.
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(a)
Suppose N is a submodule of M, let x + N be an element of M/N, and let r be a nonzero element of R; then ry = x, so
ry + N = x + N,
equivalently,
r(y + N) = x + N.
Hence M/N is divisible as well.
(b) Every element of K can be written ...
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