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    $\ZZ/p\ZZ\otimes\ZZ/q\ZZ$ for distinct primes $p$ and $q$

    P-I2MBC

    Area
    Algebra
    Topics
    Abelian Groups, Cyclic Groups, Tensor Products
    Status
    draft

    Now we’ll take the tensor product of two abelian groups, that is, \(\ZZ\dash\)modules. Take \(\ZZ/p\ZZ\) and \(\ZZ/q\ZZ\), where \(p\) and \(q\) are distinct primes. What is their tensor product?

    Solution

    1. \(\ZZ/p\otimes\ZZ/q=0\) for \(p\neq q\).

    \(a\otimes b = p(a/p)\otimes b = a\otimes pb = a\otimes0=0\) using \(p\) invertible mod \(q\); more directly \(1\otimes1 = q(1\otimes1/q)=0\) and \(p(1\otimes1)=0\), so \(\gcd(p,q)=1\) gives \(1\otimes1=0\).

    2. Q.E.D.

    1.

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