Let $A$ be a ring and $f : X \rightarrow \bigoplus_{j=1}^{n}I_{j}$ be a morphism of $A$-modules, where each $I_{j}$ is injective. If $f$ is a monomorphism, then can we conclude that there is an injective envelope $g : X \rightarrow \bigoplus_{t=1}^{m}I_{j_{t}}$? (Here $\{ j_1, \ldots, j_m \} \subset \{1, \ldots, n \}$)

If the answer is no, is there any hypothesis under $A$ which make the above statement true? For example, if $A$ is a noetherian or artin ring.

Let $A$ be an artinian ring and $f : X \rightarrow \bigoplus_{j=1}^{n}I_{j}$ be a morphism of $A$-modules, where each $I_{j}$ is injective and indecomposable. If $f$ is a monomorphism, then can we conclude that there is an injective envelope $g : X \rightarrow \bigoplus_{t=1}^{m}I_{j_{t}}$? (Here $\{ j_1, \ldots, j_m \} \subset \{1, \ldots, n \}$)