Yes: the reason is that a great deal of the logic connecting properties here uses the assumption that the ring has identity. However, we do plan to keep a list of random useful examples of rings without identity.

Please use this information!

Yes, there are! It has a few weaknesses at present. One is that it is only refreshed every once in a while, so it may take a few hours to be updated. The search should yield current results, though. Another thing is that the script that generates the entries from logic stored in the database makes some bad suggestions in edge cases. There are not many of these problematic entries, and feel free to ask a question via the suggestion form about specifics.

For a ring $R$ and an $R$ bimodule $M$, this denotes the trivial extension of $M$ by $R$. Formally it is the set $R\times M$ with addition $(r,n)+(s,m)=(r+s, n+m)$ and multiplication $(r,n)(s,m)=(rs, rm+ns)$.

For a ring $R$ and a ring $S$ which is an $R$ bimodule, this denotes the Dorroh extension of $S$ by $R$. Formally it is the set $R\times S$ with addition $(r,n)+(s,m)=(r+s, n+m)$ and multiplication $(r,n)(s,m)=(rs, rm+ns+nm)$.

Given two rings $R, S$ and an $R,S$ bimodule $M$, the set $\begin{bmatrix}R&M\\0&S\end{bmatrix}$ with matrix addition and multiplication becomes a ring. Alternatively, it could be $\begin{bmatrix}S&0\\M&R\end{bmatrix}$. We call this construction the triangular ring formed by $R,M$ and $S$.

We use $k$ to denote a countably infinite field, and $K$ for an uncountable field. $\aleph_0$ is the cardinality of the natural numbers. $\mathfrak c$ is the cardinality of the real numbers. $\Omega(n)$ is the total number of prime factors of $n$ (with repetition.)