Examples of modules such that $M\oplus N_1\simeq M\oplus N_2$, but $N_1\not\simeq N_2$?

Question

Are there any examples of modules such that $M\oplus N_1\simeq M\oplus N_2$, but $N_1\not\simeq N_2$ as modules?

I thought of taking $\bigoplus_{i\geq 0}\mathbb{Z}\oplus\mathbb{Z}\simeq\bigoplus_{i\geq 0}\mathbb{Z}\oplus\{0\}$ as $\mathbb{Z}$-modules, but $\mathbb{Z}\not\simeq\{0\}$. Is this correct? Even if it is, I felt it's kind of a cheap example, and was hoping to see something more interesting.

score 1 · Accepted Answer · answered Feb 18 '13 at 01:38

1

Yes you example is correct. Here is one which might be better. Over $\Bbb Q$ we have: $$\Bbb{R\oplus R\cong R\oplus Q}$$

Assuming the axiom of choice, anyway, this example works.

answered Feb 18 '13 at 01:38

Asaf Karagila

405,794

Thanks. Why are they isomorphic as $\mathbb{Q}$-modules? – user62738 Feb 18 '13 at 01:42
Because both are free modules of the same dimension. – Asaf Karagila Feb 18 '13 at 01:43
Oh, free modules of dimension $\mathfrak{c}$ over $\mathbb{Q}$? – user62738 Feb 18 '13 at 01:44
That is correct. This is also the reason that the additive groups of the reals and complex numbers are isomorphic n – Asaf Karagila Feb 18 '13 at 01:45

Examples of modules such that $M\oplus N_1\simeq M\oplus N_2$, but $N_1\not\simeq N_2$?

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