Cyclic Modules and Generators
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For cyclic Z-modules Zm and Zn with generators a and b, respectively, show that Zm ⊗Z Zn is isomorphic to Z(m,n) with generator a ⊗ b, where(m,n) is the greatest common divisor of m and n.
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This solution helps with a question involving cyclic modules and generators.
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Everywhere below the sign (x) is used to denote the tensor product.
Proof. It is well-known that for any abelian group A, and any natural m there is an isomorphism
h:A/mA Z_m (x)A defined as follows : h(c+mA)=u(x)c, where u is any generator of Z_m, for example u=1+mZ.
Applying this fact for A=Z_n, we obtain that there is an ...
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