All rings considered are associative. We say that the ring $R$ is an extension of the ring $I$ by the ring $T$ if $I\lhd R$ is a two-sided ideal of $R$ and $R/I\cong T$. Is there a scientific article where relationship between the ring $R$ and $I$ are investigated if $T=\mathbb{Z}$? In particular, I am interested what are the necessary and sufficient conditions for ring $I$ to every extension of $I$ by $\mathbb{Z}$ is commutative. It is easy to see that the commutativity of $I$ is a necessary but not a sufficient condition (example of non-commutative ring $R=\{\left[\begin{array}{cc} a & b\\0 &a\end{array}\right]\colon a,b\in \mathbb{Z}\}$ which is an extension of the commutative (with zero multiplication) ring $I=\{\left[\begin{array}{cc} 0 & b\\0 &0\end{array}\right]\colon b\in \mathbb{Z}\}$ by the ring of integers $R/I\cong \mathbb{Z}$).
An extra (more general) problem: When the extension of commutative ring by a commutative ring is commutative? I will be very grateful for indicating the works in which you can read about these issues.