My book says the following:
Let $M $ and $N $ be modules. Then $M\times N $ is not a module.
I don't understand this statement. It seems to satisfy all properties of modules.
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1Are we talking about modules over a commutative ring or over a non-commutative ring? Because in the non-commutative case, the product of a right module and a left module doesn't necessarily have a well-defined module structure. – Arthur Oct 21 '15 at 12:38
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@Arthur- I was also thinking scalar product might not be well defined. Let us take $(m, n)-(m, n) $. Is this equal to $(m, n)+(-m, n) $? – Oct 21 '15 at 12:40
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I don't understand the statement either. If you take the product of two left modules, you get a left module - the category of left modules over a fixed ring, commutative or otherwise, forms an abelian category, which includes finite products. – Dustan Levenstein Oct 21 '15 at 12:45
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You can make it a module with proper definitions but as pointed out, the general case is not necciserly so – Zelos Malum Oct 21 '15 at 12:52
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Which book are you talking about? More context might help resolve what the book means. – Matthew Towers Oct 21 '15 at 13:03
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Dear @MattSamuel : that does not work for unital modules, at least. You would have $(2m,n)=2(m,n)=(m,n)+(m,n)$, and subtracting from both sides yields $(m,0)=(m,n)$, an absurdity if $n\neq 0$. Regards – rschwieb Oct 24 '15 at 03:25