Let $R$ be a ring with unity and $I$ be an ideal of $R$. Given a set $\{R_\lambda:\lambda\in\Lambda\}$ of rings. We define the direct product $C = \prod_\limits{\lambda\in\Lambda} R_\lambda$ as a ring $C$ such that
i) to each $\lambda\in\Lambda$ there exists a surjection $\pi_\lambda:C\to R_\lambda$,
ii) if $B$ is a ring and $f_\lambda: B\to R_\lambda, \lambda\in\Lambda$, system of homomorphisms then there exists a unique homomorphism $f: B\to C$ such that $\pi f=f_\lambda$ for all $\lambda\in\Lambda$.
Using that definition, is it always true that $$I\times R/I=R~?$$
I have a feeling that the product is just $I\times R/I\cong R~$ because $R/I$ is just a set of cosets, yet $I$ contains $R$ elements.
Since $R\cong I\times R/I$, we have $I$ is a finitely generated ideal of $R$, also, since $R$ has an identity, $I$ has an identity, say $x$. Then $ix=xi=i$ for all $i\in I$. Thus $I^2=I$. Hence, $I$ is an idempotent finitely generated ideal of $R$. Thus there is a central idempotent $e$ of $R$ such that $I=Re$.
The converse is also true since for a central idempotent $e$ of $R$, we have $R\cong Re\times R(1-e)\cong Re\times R/Re$.
Therefore, $R\cong I\times R/I$ if and only if there is a central idempotent $e$ of $R$ such that $I=Re$.