Let \(R\) be a commutative ring and let \(M\) be an \(R\)-module. An element \(x \in M\) is a torsion element if there exists a non-zero \(r \in R\) such that \(rx = 0\) (or equivalently, \(\operatorname{Ann}_R(x) \neq (0)\)). Let \(\operatorname{Tor}(M) = \{x \in M : \operatorname{Ann}_R(x) \neq (0)\}\) denote the set of torsion elements of \(M\).
Prove that if \(R\) is an integral domain, then \(\operatorname{Tor}(M)\) is an \(R\)-submodule of \(M\).
Prove that the quotient module \(M / \operatorname{Tor}(M)\) is torsion-free.
Show by example that \(\operatorname{Tor}(M)\) need not be a submodule if \(R\) has zero divisors.
Solution
1. Prove that \(\operatorname{Tor}(M)\) is a submodule when \(R\) is an integral domain:
1.1. For \(x = 0 \in M\), \(1 \cdot 0 = 0\) with \(1 \neq 0\) in \(R\), so \(0 \in \operatorname{Tor}(M)\).
\(1 \in \operatorname{Ann}_R(0)\).
1.2. Let \(x, y \in \operatorname{Tor}(M)\).There exist non-zero \(r_1, r_2 \in R \setminus \{0\}\) such that \(r_1 x = 0\) and \(r_2 y = 0\).
definition of torsion elements.
1.3. Since \(R\) is an integral domain and \(r_1 \neq 0, r_2 \neq 0\), the product \(r_1 r_2 \neq 0\) in \(R\).
integral domains have no zero divisors.
1.4. Compute \((r_1 r_2)(x + y)\):\[ (r_1 r_2)(x + y) = r_2(r_1 x) + r_1(r_2 y) = r_2(0) + r_1(0) = 0. \]
commutativity of \(R\) and module axioms.
1.5. Since \(r_1 r_2 \neq 0\), \(x + y \in \operatorname{Tor}(M)\).
1.4.
1.6. For any \(c \in R\) and \(x \in \operatorname{Tor}(M)\) with \(r_1 x = 0\) (\(r_1 \neq 0\)):
\[ r_1(cx) = c(r_1 x) = c(0) = 0. \] Thus \(cx \in \operatorname{Tor}(M)\).
commutativity of \(R\).
1.7. Therefore \(\operatorname{Tor}(M)\) is closed under addition and scalar multiplication, hence an \(R\)-submodule of \(M\).
subspace/submodule criterion.
2. Prove that \(M / \operatorname{Tor}(M)\) is torsion-free:
2.1. Let \(\bar{x} = x + \operatorname{Tor}(M) \in M / \operatorname{Tor}(M)\) be a torsion element.
setup.
2.2. Then there exists some non-zero \(s \in R \setminus \{0\}\) such that \(s \bar{x} = \bar{0}\) in \(M / \operatorname{Tor}(M)\).
definition of torsion in quotient module.
2.3. This means \(sx \in \operatorname{Tor}(M)\).
definition of cosets in quotient module.
2.4. Since \(sx \in \operatorname{Tor}(M)\), there exists some non-zero \(t \in R \setminus \{0\}\) such that \(t(sx) = 0\).
definition of \(\operatorname{Tor}(M)\).
2.5. Rewrite \(t(sx) = (ts)x = 0\).Since \(R\) is an integral domain and \(t \neq 0, s \neq 0\), the product \(ts \neq 0\).
integral domain property.
2.6. Since \((ts)x = 0\) with \(ts \neq 0\), \(x \in \operatorname{Tor}(M)\).
definition of \(\operatorname{Tor}(M)\).
2.7. Thus \(\bar{x} = x + \operatorname{Tor}(M) = \bar{0}\) in \(M / \operatorname{Tor}(M)\).
\(x \in \operatorname{Tor}(M) \implies x + \operatorname{Tor}(M) = \operatorname{Tor}(M)\).
2.8. Hence \(\operatorname{Tor}(M / \operatorname{Tor}(M)) = \{\bar{0}\}\), so \(M / \operatorname{Tor}(M)\) is torsion-free.
1 through 7.
3. Counterexample when \(R\) is not an integral domain:
3.1. Let \(R = \mathbb{Z}/6\mathbb{Z}\) and \(M = R = \mathbb{Z}/6\mathbb{Z}\) as a module over itself.
\(R\) has zero divisors since \(2 \cdot 3 = 0\).
3.2. The element \(x = [2]\) is torsion because \(3 \cdot [2] = [0]\) with \([3] \neq [0]\) in \(R\).
\(3 \cdot 2 = 6 \equiv 0 \pmod 6\).
3.3. The element \(y = [3]\) is torsion because \(2 \cdot [3] = [0]\) with \([2] \neq [0]\) in \(R\).
\(2 \cdot 3 = 6 \equiv 0 \pmod 6\).
3.4. However, their sum \(x + y = [5]\) satisfies \(\operatorname{Ann}_R([5]) = \{[0]\}\) because \(\gcd(5, 6) = 1\) (\([5]\) is a unit in \(\mathbb{Z}/6\mathbb{Z}\)). Thus \(x + y = [5] \notin \operatorname{Tor}(M)\).
unit elements have trivial annihilators.
3.5. Therefore \(\operatorname{Tor}(M)\) is not closed under addition when \(R\) has zero divisors.
2, 3, and 4.
4. Conclusion: \(\operatorname{Tor}(M)\) is a submodule and \(M/\operatorname{Tor}(M)\) is torsion-free over any integral domain, but closure under addition fails in the presence of zero divisors.
1, 2, and 3.
Q.E.D.