Hi I am looking for a result about the following elementary question:

Suppose $A$ is an algebra and its center $C$ is an Artinian ring. It is well known that $C$ can be decomposed into a product of local Artinian rings: $C \cong C_1 \times \cdots \times C_t $. Would this decomposition lead to a certain decomposition of the algebra $A$? If so, I'd to konw a reference about this. Thanks