Is not it Null?

Let $\Lambda$ denote the null string.

$A$ and $B$ are two languages over the alphabet $\Sigma$ such that neither of them is empty nor $\Sigma^*$ .

True or False?

$AB=A \implies B= \{\Lambda\}$

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Context : Theory of Computation and Languages

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Thanks to: Agnishom Chattopadhyay