# Center of algebra of functions

Let $A$ be an algebra over a field (not necessarily commutative) with multiplication $m$ and unit $1_A$, and $X$ be a finite set. The set of functions from $X$ to $A$, $Fun(X,A)$, can be given an algebra structure with the obvious pointwise definitions: $\forall x \in X, (f+g)(x):=f(x)+g(x), (\lambda f)(x):=\lambda f(x), (fg)(x):=f(x)g(x)$. Clearly the center $Z=\{f \in Fun(X,A): fg=gf, \forall g \in Fun(X,A)\}$  of $Fun(X,A)$ is the whole $Fun(X,A)$ if $A$ is commutative (since $Fun(X,A)$ is also commutative), but what is the center if $A$ is not commutative?

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