Let \(R\) be a commutative ring with identity \(1 \ne 0\), and let \(M\) be an \(R\)-module. Recall that the set of torsion elements of \(M\) is defined by \[ \operatorname{tor}(M) = \{m \in M \mid \exists r \in R \setminus \{0\} \text{ such that } r m = 0\}. \]
Prove that if \(R\) is an integral domain, then \(\operatorname{tor}(M)\) is an \(R\)-submodule of \(M\).
Give an example where \(\operatorname{tor}(M)\) is not an \(R\)-submodule of \(M\).
If \(R\) has zero-divisors, prove that every non-zero \(R\)-module \(M \ne \{0\}\) contains at least one non-zero torsion element.
Solution
Goal: Prove that \(\operatorname{tor}(M)\) is a submodule over integral domains in (a), provide a counterexample over non-domains in (b), and prove the existence of non-zero torsion elements over rings with zero-divisors in (c).
1. Part (a): \(\operatorname{tor}(M)\) is an \(R\)-submodule when \(R\) is an integral domain.
1.1. Non-empty: \(0 \in \operatorname{tor}(M)\) since \(1 \cdot 0 = 0\) and \(1 \ne 0\) in \(R\).
1.2. Closure under scalar multiplication:- Let \(m \in \operatorname{tor}(M)\) and \(r \in R\).- There exists \(s \in R \setminus \{0\}\) such that \(s m = 0\).- Then \(s (r m) = r (s m) = r \cdot 0 = 0\) by commutativity of \(R\).- Since \(s \ne 0\), \(r m \in \operatorname{tor}(M)\).
1.3. Closure under addition:- Let \(m_1, m_2 \in \operatorname{tor}(M)\).- There exist \(s_1, s_2 \in R \setminus \{0\}\) such that \(s_1 m_1 = 0\) and \(s_2 m_2 = 0\).- Multiply the sum \(m_1 + m_2\) by \(s_1 s_2 \in R\):\[(s_1 s_2)(m_1 + m_2) = s_2 (s_1 m_1) + s_1 (s_2 m_2) = s_2 \cdot 0 + s_1 \cdot 0 = 0.\]- Since \(R\) is an integral domain and \(s_1 \ne 0, s_2 \ne 0\), the product \(s_1 s_2 \ne 0\).- Thus \(m_1 + m_2 \in \operatorname{tor}(M)\).
1.4. Therefore \(\operatorname{tor}(M)\) is an \(R\)-submodule of \(M\).
2. Part (b): Counterexample where \(\operatorname{tor}(M)\) is not a submodule.
2.1. Let \(R = \mathbb{Z}/6\mathbb{Z}\) and consider \(M = R = \mathbb{Z}/6\mathbb{Z}\) as a regular module over itself.
2.2. The element \(\bar{2} \in \mathbb{Z}/6\mathbb{Z}\) is torsion: \(3 \cdot \bar{2} = \bar{6} = \bar{0}\) with \(\bar{3} \ne \bar{0}\), so \(\bar{2} \in \operatorname{tor}(M)\).
2.3. The element \(\bar{3} \in \mathbb{Z}/6\mathbb{Z}\) is torsion: \(2 \cdot \bar{3} = \bar{6} = \bar{0}\) with \(\bar{2} \ne \bar{0}\), so \(\bar{3} \in \operatorname{tor}(M)\).
2.4. The sum is \(\bar{2} + \bar{3} = \bar{5}\).
2.5. Since \(\gcd(5, 6) = 1\), \(\bar{5}\) is a unit in \(\mathbb{Z}/6\mathbb{Z}\) (\(\bar{5}^2 = \bar{25} = \bar{1}\)).
2.6. For any \(r \in \mathbb{Z}/6\mathbb{Z}\), \(r \cdot \bar{5} = \bar{0} \implies r = r \cdot \bar{5} \cdot \bar{5} = \bar{0} \cdot \bar{5} = \bar{0}\).
2.7. Thus \(\bar{5} \notin \operatorname{tor}(M)\), showing that \(\operatorname{tor}(M)\) is not closed under addition.
3. Part (c): Existence of non-zero torsion elements when \(R\) has zero-divisors.
3.1. Since \(R\) has zero-divisors, there exist non-zero elements \(a, b \in R \setminus \{0\}\) such that \(a b = 0\).
3.2. Let \(M \ne \{0\}\) be any non-zero \(R\)-module. Choose a non-zero element \(m \in M \setminus \{0\}\).
3.3. Consider the element \(b m \in M\):
3.4. Case 1 (\(b m = 0\)):- The element \(m \in M\) is non-zero, and \(b m = 0\) with \(b \in R \setminus \{0\}\).- Thus \(m\) is a non-zero torsion element of \(M\).
3.5. Case 2 (\(b m \ne 0\)):- Define \(w = b m \in M \setminus \{0\}\).- Compute \(a w = a (b m) = (a b) m = 0 \cdot m = 0\).- Since \(a \in R \setminus \{0\}\) and \(w \ne 0\), \(w = b m\) is a non-zero torsion element of \(M\).
3.6. In both cases, \(M\) contains at least one non-zero torsion element.
4. Conclusion:
$\operatorname{tor}(M)$ is a submodule over integral domains, fails to be a submodule in general rings, and non-zero modules over rings with zero-divisors always contain non-zero torsion elements.