Injective ring homomorphism between finite direct product of rings

Question

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 ... )

Answer 1

Suppose $\alpha:R^n\times R\to R^n$ is an injective ring homomorphism. Extend the codomain to get a (non-unital) endomorphism $\beta:R^n\times R\to R^n\times R$, where $\beta(x)=(\alpha(x),0)$.

Consider the idempotent element $e_0=(0,1)\in R^n\times R$ and its images $e_n=\beta^n(e_0)$ under powers of $\beta$. Then $\{e_n\vert n\geq0\}$ is an infinite set of orthogonal idempotents ($e_ie_j=0$ for $i\neq j$) since $e_0\beta(e_0)=0$.

The ideal generated by $\{e_n\vert n\geq0\}$ is infinitely generated, so $R^n\times R$ is not noetherian. Since a finite direct product of noetherian rings is noetherian, $R$ is not noetherian.

Answer 2

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.