If $A\oplus B\approx A$ then $B=0$?

Question

Suppose $A,B$ abelian groups, such that $A\oplus B\approx A$, can I conclude that $B=0$?

If it's true, is there any hint how to prove it?

You need more assumption than that, assuming $\oplus$ means direct sum. It would hold for finite abelian groups. — hardmath, Jun 22 '19 at 18:45

score 3 · Accepted Answer · answered Jun 22 '19 at 18:44

3

No; consider $A=\Bbb{Z}^{\Bbb{N}}=\bigoplus_{n\in\Bbb{N}}\Bbb{Z}$.

answered Jun 22 '19 at 18:44

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