Let \(R\) be a commutative ring and \(M\) an \(R-\)module.
Define what a torsion element of \(M\) is .
Given an example of a ring \(R\) and a cyclic \(R-\)module \(M\) such that \(M\) is infinite and \(M\) contains a nontrivial torsion element \(m\). Justify why \(m\) is torsion.
Show that if \(R\) is a domain, then the subset of elements of \(M\) that are torsion is an \(R-\)submodule of \(M\). Make explicit where the hypothesis that \(R\) is a domain is used.
Solution
1. An element \(m\in M\) is torsion if there exists a nonzero \(r\in R\) such that\[ rm=0. \]
The zero element is always torsion; a nonzero torsion element is one annihilated by some nonzero scalar.
2. There is an infinite cyclic module with a nonzero torsion element.
Take \[ R=\ZZ\times\ZZ, \qquad M=R \] with the regular \(R\)-module structure. Then \(M\) is cyclic, generated by \(1_R=(1,1)\), and its underlying set is infinite. The nonzero element \[ m=(1,0) \] is torsion because the nonzero scalar \[ r=(0,1) \] satisfies \[ rm=(0,1)(1,0)=(0,0). \]
Now assume for the remaining parts that \(R\) is a domain, and let \[ T(M)=\{m\in M:m\text{ is torsion}\}. \]
3. The zero element lies in \(T(M)\).
For any nonzero \(r\in R\), \[ r0=0. \]
4. If \(m,n\in T(M)\), then \(m+n\in T(M)\).
Choose nonzero \(a,b\in R\) with \[ am=0, \qquad bn=0. \] Then \[ (ab)(m+n)=b(am)+a(bn)=0. \] Because \(R\) is a domain and \(a,b\ne0\), one has \(ab\ne0\). Hence \(m+n\) is torsion. This is exactly where the domain hypothesis is used.
5. If \(m\in T(M)\) and \(r\in R\), then \(rm\in T(M)\).
Choose nonzero \(a\in R\) with \(am=0\). Then \[ a(rm)=r(am)=0. \] Since the same nonzero scalar \(a\) annihilates \(rm\), it is torsion.
6. Therefore \(T(M)\) is an \(R\)-submodule of \(M\) when \(R\) is a domain.
By <1>3–<1>5, \(T(M)\) contains \(0\) and is closed under addition and scalar multiplication.