I know I have met the following construction somewhere, but I cannot remember where. Let $A$ be a (unital associative) ring, and let $N$ be an $A$-$A$ bimodule. On the product set $A\times N$ we define multiplication by \begin{equation*} (a,m)(b,n) := (ab,\,an+mb)~. \end{equation*} The set $A\times N$ equipped with this multiplication is a (unital associative) ring with the two-sided ideal $0\times N$ whose product with itself is $0$. The ideal $0\times N$ is the kernel of the surjective homomorphism of rings $A\times N\to A : (a,n)\mapsto a$. If $A$ is Jacobson-semisimple, then $0\times N$ is the Jacobson radical of the ring $A\times N$.

Does this construction of the ring $A\times N$ from a ring $A$ and an $A$-$A$ bimodule $N$ have a name, and perhaps an established notation? The notation $A\times N$ is misleading since it suggests a direct product of rings, which it is not.

This is not part of the question, it is a request for help with editing questions and answers in mathoverflow. In mathoverflow I avoid asking questions and giving answers because the preview of mathematical formulas does not work: in the preview the formulas are shown precisely as I type them, with dollars and all. On one previous occasion I incurred a wrath of a couple of users because I repeatedly posted the answer I was typing, since there was no other way I could see what I was doing. For a while I thought I found the cause of the problem: replacing http in the URL with https seemed to help, since math preview worked -- alas, only for a couple of lines, then the math preview died on me. Since then I shunned mathoverflow like it was poisoned. I did look for an explanation and an advice what to do. The problem is not unknown, it seems. But nothing resembling an advice, which I found on the 'soft' part of mathoverflow, really worked. Can you help me here? I would very much like to be more active on mathoverflow, but this problem makes the site practically inaccessible to me.