finite abelian groups, tensor product

**riemannsph12** · May 13th 2009, 05:44 PM

If $A$ and $B$ are finite abelian groups of relatively prime orders, prove that $A \otimes_{\mathbb{Z}} B=0$ . If $p$ is prime and $r>s$ , prove $(\mathbb{Z}/p^r\mathbb{Z})\otimes_{\mathbb{Z}}(\mathbb{Z}/p^s\mathbb{Z})=\mathbb{Z}/p^s\mathbb{Z}$ .

Any hints would be great (I haven't gotten far with this one!). Thanks in advance.

**NonCommAlg** · May 13th 2009, 06:27 PM

Originally Posted by riemannsph12

If $A$ and $B$ are finite abelian groups of relatively prime orders, prove that $A \otimes_{\mathbb{Z}} B=0$ .

let $|A|=n, \ |B|=m,$ and $a \in A, \ b\in B.$ since $\gcd(n,m)=1,$ there exist $r,s \in \mathbb{Z}$ such that $rn+sm=1.$ thus: $a \otimes b=[(rn+sm)a] \otimes b = (sma) \otimes b = (sa) \otimes mb = (sa) \otimes 0 = 0.$

If $p$ is prime and $r>s$ , prove $(\mathbb{Z}/p^r\mathbb{Z})\otimes_{\mathbb{Z}}(\mathbb{Z}/p^s\mathbb{Z})=\mathbb{Z}/p^s\mathbb{Z}$ .

see my post in http://www.mathhelpforum.com/math-he...tensor-zn.html for a general form of your question.

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