A noetherian ring is self-injective if and only if all projectives are injective

Question

Let $R$ be a left and right noetherian ring. I try to prove that the following two statements are equivalent.

1. $R$ is injective as a left $R$-module

2. All projective left $R$-modules are injective

In my class, left (right) noetherian ring is defined in such the way that any ACC on left(right) ideal stop at some point. Also, $A$ is injective if for any injective $f:X \to Y$ and homomorphism $g:X \to A$ there exist $h: Y \to A$ such that f$\circ h=g$.

I still have no idea how to apply ACC condition on those homomorphism. Could you help? Any hints or answers would be really appreciated.

Answer 1

The implication $(2)\implies(1)$ is obvious, because $R$ is projective as $R$-module.

The other direction is usually proved indirectly; the theorem to use is the following:

The following conditions on the ring $R$ are equivalent:

1. $R$ is left noetherian
2. Every direct sum of injective left $R$-modules is injective

(you need only one implication of this theorem).

Recall also that a summand of an injective (resp. projective) module is injective (resp. projective) and that a projective module is a summand of a free module.