Injective ring homomorphism between finite direct product of rings All rings below are commutative with unity.
Let $R$ be a Noetherian ring such that  there is an injective ring homomorphism from $R^m$ to $R^n$ , then is it true that $m\le n $ ? If not true in general then can we impose any condition (apart from Artinian) on $R$ to make the statement true ?
( NOTE : I know the statement to be true for any ring if we consider injective module homomorphism ... )

You can easily reduce to the Artinian case by localizing at a minimal prime. Localization is flat, so you still get an injective homomorphism and the ring is now Artinian. 
