Definition and basic properties of the set of torsion elements in a module

Let R be a ring with 1 \neq 0 and let M be a left R-module. An element m \in M is called torsion if there exists r \in R such that r \cdot m = 0. We define \mathsf{Tor}_R(M) = \{ m \in M \ |\ r \cdot m = 0\ \mathrm{for\ some\ nonzero}\ r \in R \}.

  1. Prove that if R is an integral domain, then \mathsf{Tor}(M) is a submodule of M.
  2. Show by an example that if R is not an integral domain, then \mathsf{Tor}(M) need not be a submodule.
  3. Prove that if R has zero divisors, then every nontrivial R module has nonzero torsion elements.

  1. We use the submodule criterion. Note that 1 \cdot 0 = 0 in M, so that 0 \in \mathsf{Tor}(M). Now suppose a,b \in \mathsf{Tor}(M) and r \in R. There exist nonzero u,v \in R such that u \cdot a = v \cdot b = 0. Since R is an integral domain, uv \neq 0. Now uv \cdot (a+r \cdot b) = v \cdot (u \cdot a) + r \cdot (v \cdot b) = 0 (since R is commutative), so that a+r \cdot b \in \mathsf{Tor}(M). Thus \mathsf{Tor}(M) is a submodule of M.
  2. Consider the ring R = \mathbb{Z}/(6), and consider M = R as a left module over itself. Note that neither \overline{2} nor \overline{3} are zero in R, but that \overline{3} \cdot \overline{2} = \overline{2} \cdot \overline{3} = 0. Thus \overline{2}, \overline{3} \in \mathsf{Tor}(M). However, \overline{2} + \overline{3} = \overline{5}, and r \cdot \overline{5} = 0 only if r = 0. Thus \mathsf{Tor}(M) is not a submodule of M, as it is not closed under addition.
  3. Suppose R has zero divisors and let M be a left R-module. Suppose by way of contradiction that \mathsf{Tor}(M) = 0. Now let r,s \in R be nonzero such that rs = 0, and let m \in M be nonzero. Now r \cdot (s \cdot m) = rs \cdot m = 0 \cdot m = 0, so that s \cdot m is torsion. Thus s \cdot m = 0, and so m = 0, a contradiction. Thus every nontrivial R-module contains nonzero torsion elements.
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