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Problem 2. An element m of the R-module M is called a torsion element if rm = 0 for somenonzero element r € R. The set of torsion elements is denoted Tor(M) = {m € M | rm =0 for some nonzero r = R}. (a) Prove that if R is an integral domain then Tor(M) is a submodule of M (called thetorsion submodule of M). (b) Give an example of a ring R and an R-module M such that Tor(M) is not a submodule. (c) If R has zero divisors show that every nonzero R-module has nonzero torsion elements.

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