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Nov 1, 2017 at 1:04 comment added Mohan The one situation where this is obvious is if $\alpha\beta$ is a non-zero divisor in $A$. For injectivity on the left, you need just that $\beta$ is a non-zero divisor in $A$. If that is not the case, in general this can be false without further conditions.
Nov 1, 2017 at 0:16 history edited D. Bogiokas CC BY-SA 3.0
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Nov 1, 2017 at 0:03 history asked D. Bogiokas CC BY-SA 3.0