Given the central ring homomorphism \(\mathit {1}\), define
\(r \bullet x=\mathit {1}(r)*x\). The ring laws and the preservation of addition,
multiplication and identity by \(\mathit {1}\) give
\[
\begin {aligned}
r \bullet (x+y)&=\mathit {1}(r)*x+\mathit {1}(r)*y,\\
(r+s)\bullet x&=\mathit {1}(r)*x+\mathit {1}(s)*x,\\
r\bullet (s\bullet x)&=(\mathit {1}(r)*\mathit {1}(s))*x=(r*s)\bullet x,\\
1_R\bullet x&=1_A*x=x.
\end {aligned}
\]
Thus the action defines an \(R\)-module. Associativity gives
\(r\bullet (x*y)=(r\bullet x)*y\); centrality of \(\mathit {1}(r)\) then gives
\((r\bullet x)*y=x*(r\bullet y)\).
Conversely, given the module formulation, put
\(\mathit {1}(r)=r\bullet 1_A\). The module axioms give
\(\mathit {1}(r+s)=\mathit {1}(r)+\mathit {1}(s)\) and \(\mathit {1}(1_R)=1_A\).
They also give \(\mathit {1}(0_R)=0_A\), by cancellation in
\(0_R\bullet 1_A=(0_R\bullet 1_A)+(0_R\bullet 1_A)\).
Compatibility with multiplication implies
\[
\begin {aligned}
\mathit {1}(r)*\mathit {1}(s)
&=(r\bullet 1_A)*(s\bullet 1_A)\\
&=r\bullet (1_A*(s\bullet 1_A))\\
&=(r*s)\bullet 1_A=\mathit {1}(r*s),\\
\mathit {1}(r)*x&=r\bullet x=x*\mathit {1}(r).
\end {aligned}
\]
Hence \(\mathit {1}\) is a central, identity-preserving ring homomorphism,
and its induced action is the original one.
Starting with a central ring homomorphism instead recovers it because
\(\mathit {1}(r)*1_A=\mathit {1}(r)\).