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At first glance, the rank seems to be $2$. The basis elements seem to be $(0,1), (1,0)$.

However, how is scalar multiplication defined? Is $a(p,q)=(ap,q)=(p,aq)$? So if the module under consideration is $\Bbb{Z}_2\oplus \Bbb{Z}_3$, is $5(1,2)=(5,2)=(1,2)=(1,10)=(1,1)$?

freebird
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  • What do you mean by $\mathbb{Z}_m$? Is it the finite cyclic group with $m$ elements? If so, the rank is obviously $0$. – egreg Apr 25 '16 at 20:20
  • @egreg- Yes I mean exactly that. Could you explain why? I suppose I do not fully understand what the rank of a module. Isn't it supposed to be the (smallest) number of elements in a generating set of the module? – freebird Apr 25 '16 at 20:21
  • $(ap,q)=(p,aq)$ would hold for a tensor product of modules, but $\oplus$ indicates a direct sum, which really is just the Cartesian product with appropriate operations on it -- in particular, $a(p,q)=(ap,aq)$. – hmakholm left over Monica Apr 25 '16 at 20:25

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