Given a ring R how to obtain a ring $R^1$ with an identity?

Question

Let $R$ be a ring (associative, but need not have an identity). The usual extension (https://en.wikipedia.org/wiki/Ring_extension) of R by $\mathbb{Z}$ gives a ring $R^1$ with an identity element.

How to construct this usual extension?

Thanks in advance.

Answer 1

To obtain an extension of $R$ with identity $e$, consider $R^1=R[e]=\{r+me/r\in R, m\in \mathbb{Z}\}$