Modules over Noetherian RIngs

Question

Let $A$ be an associative Noetherian algebra (not necessarily commutative) over $\mathbb{C}$ such that every irreducible $A$-module is an injective $A$-module. Can we conclude that every $A$-module is a finite direct sum of irreducible $A$-modules? I think we can write any $A$-module as a direct sum (possibly infinite) of irreducible $R$-modules and for that one does nor require $A$ to be either associative or Noetherian, but my main aim is to show (if it is possible) that in fact it is a finite sum.Thanks for any help.