# FAQs

Do all the rings here have identity?
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.
How can I cite this site in a work?
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.)