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According to the definition, the productions of a right linear grammar should have the form of $A\to xB$ or $A\to x$, does $A\to B$ or $A\to xy$ count as productions of a right linear grammar? $A\to B$ can be written as $A\to \epsilon B$ though...

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In your definition, $x$ must be a terminal symbol. In particular, it cannot be $\epsilon$, which is the empty string. The productions $A \to B$ and $A \to xy$ don't conform to the specifications you give, hence they don't belong in a right linear grammar according to the definition you provide.

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