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Thread: Zm tensor Zn

  1. #1
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    Zm tensor Zn

    This is a standard problem I've seen, but I am new to tensors. If I could see the proof I think it would help me greatly to compute tensors in general.

    The problem statement here is from Hungerford GTM section IV.5.2

    Show $\displaystyle Z_m \otimes Z_n \cong Z_c$ where c is gcd(m,n).

    (everything is regarded as an add. abelian group-- the tensor is over Z)

    A lead question is to show $\displaystyle A \otimes Z_m \cong A/mA$, which I think I managed with the map f sending $\displaystyle a \otimes (n+mZ) \mapsto (an+mA)$ showing the kernel was 0 and was surjective. For what it's worth, as a bijection this implies elements of $\displaystyle A \otimes Z_m$ are required to have the form simply $\displaystyle a \otimes 1$-- whether or not this will help with the problem stated above has not been something I can reconcile on my own.

    Any insights would be appreciated greatly.
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  2. #2
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    Quote Originally Posted by gosualite View Post
    This is a standard problem I've seen, but I am new to tensors. If I could see the proof I think it would help me greatly to compute tensors in general.

    The problem statement here is from Hungerford GTM section IV.5.2

    Show $\displaystyle Z_m \otimes Z_n \cong Z_c$ where c is gcd(m,n).

    (everything is regarded as an add. abelian group-- the tensor is over Z)

    A lead question is to show $\displaystyle A \otimes Z_m \cong A/mA$, which I think I managed with the map f sending $\displaystyle a \otimes (n+mZ) \mapsto (an+mA)$ showing the kernel was 0 and was surjective. For what it's worth, as a bijection this implies elements of $\displaystyle A \otimes Z_m$ are required to have the form simply $\displaystyle a \otimes 1$-- whether or not this will help with the problem stated above has not been something I can reconcile on my own.

    Any insights would be appreciated greatly.
    To show $\displaystyle A \otimes Z_m \cong A/mA$, the first isomorphism theorem can be used.

    First define a surgective map f such that

    $\displaystyle f:A \otimes Z_m \rightarrow A$ given by $\displaystyle (a,x) \mapsto a $, where $\displaystyle a \in A , x \in Z_m$.

    f is clearly surgective. Now, you need to find a kernel of f.

    Since $\displaystyle ma \otimes x = a \otimes mx = a \otimes 0 = 0 $, mA is a kernel of f.

    To show $\displaystyle Z_m \otimes Z_n \cong Z_c$, you can use the above result and replace A with $\displaystyle Z_n$.

    Since $\displaystyle Z_m \otimes Z_n \cong Z_n/mZ_n$, you remain to show that $\displaystyle Z_n/mZ_n \cong Z_c$. In a similar manner with the above, you can find a surgective map and its kernel and apply the first isomorphism theorem.
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  3. #3
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    Quote Originally Posted by aliceinwonderland View Post
    To show $\displaystyle A \otimes Z_m \cong A/mA$, the first isomorphism theorem can be used.

    First define a surgective map f such that

    $\displaystyle f:A \otimes Z_m \rightarrow A$ given by $\displaystyle (a,x) \mapsto a $, where $\displaystyle a \in A , x \in Z_m$.

    f is clearly surgective. Now, you need to find a kernel of f.

    Since $\displaystyle ma \otimes x = a \otimes mx = a \otimes 0 = 0 $, mA is a kernel of f.
    that's not correct! here's how to do it: the map $\displaystyle f_0: A \times \mathbb{Z}_m \longrightarrow A/mA$ defined by $\displaystyle f_0(a,k)=ka + mA$ is $\displaystyle \mathbb{Z}$ bilinear and thus it induces $\displaystyle f: A \otimes_{\mathbb{Z}} \mathbb{Z}_m \longrightarrow A/mA,$ which maps a generator

    $\displaystyle a \otimes k$ to $\displaystyle ka + mA.$ now define $\displaystyle g:A/mA \longrightarrow A \otimes_{\mathbb{Z}} \mathbb{Z}_m$ by $\displaystyle g(a + mA)=a \otimes 1.$ it's clear that $\displaystyle g$ is a $\displaystyle \mathbb{Z}$ homomorphism. it's well-defined because if $\displaystyle a + mA=0,$ then $\displaystyle a=ma',$ and therefore

    $\displaystyle a \otimes 1 = (ma') \otimes 1 = a' \otimes m=0.$ now we have $\displaystyle fg(a+mA)=f(a \otimes 1)=a+mA.$ hence $\displaystyle fg= \text{id}_{A/mA}$ and $\displaystyle gf(a \otimes k)=g(ka + mA)=(ka) \otimes 1=a \otimes k,$ i.e. $\displaystyle gf= \text{id}_{A \otimes \mathbb{Z}}. \ \Box$

    ... you remain to show that $\displaystyle Z_n/mZ_n \cong Z_c$.
    we have: $\displaystyle m(\mathbb{Z}/n\mathbb{Z})=(m\mathbb{Z}+n\mathbb{Z})/n \mathbb{Z}=c \mathbb{Z}/n \mathbb{Z}.$ thus $\displaystyle \mathbb{Z}_n / m \mathbb{Z}_n = (\mathbb{Z}/n \mathbb{Z})/(c \mathbb{Z}/n \mathbb{Z}) \cong \mathbb{Z}/c \mathbb{Z}=\mathbb{Z}_c.$


    Notation: you should use $\displaystyle \mathbb{Z}/n$ or $\displaystyle \mathbb{Z}/n \mathbb{Z}$ instead of $\displaystyle \mathbb{Z}_n.$

    Remark: a trivial generalization of this problem is that for any unitary commutative ring $\displaystyle R,$ ideals $\displaystyle I,J$ and $\displaystyle R$ module $\displaystyle A$ we have: $\displaystyle \frac{R}{I} \otimes_R A \cong \frac{A}{IA},$ and $\displaystyle \frac{R}{I} \otimes_R \frac{R}{J} \cong \frac{R}{I+J}.$


    by the way i'm not completely back yet! i'll try to help whenever i can (unfortunately not as often as before! don't blame me, blame my advisor!)
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  4. #4
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    Thank you both. It was this one fact:

    Quote Originally Posted by NonCommAlg View Post
    we have: $\displaystyle m(\mathbb{Z}/n\mathbb{Z})=(m\mathbb{Z}+n\mathbb{Z})/n \mathbb{Z}=c \mathbb{Z}/n \mathbb{Z}.$
    that gcd is the least linear combination, that had eluded my attempts to prove this. I appreciate both replies greatly.
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