zero ring

《Abstract Algebra》 · 第 224 页 · Chapter 7，Section 7.1，Example (1)

In particular, if $R=\{0\}$ is the trivial group, the resulting ring $R$ is called the zero ring, denoted $R=0$. Except for the zero ring, a trivial ring does not contain an identity ($R=0$ is the only ring where $1=0$; we shall often exclude this ring by imposing the condition $1\ne 0$).

《The foundations of geometry》 · 第 43—44 页 · Chapter II，Section 3，Example 3.8

One simple example that defies our experience with numbers is the ring consisting of a single element, say $r$, with operations $r+r:=r$, $r\cdot r:=r$. You can scan quickly the ring axioms and verify that they are all trivially satisfied; and indeed, the corresponding ring is called the 'trivial' ring. In the trivial ring $0_R$ and $1_R$ both equal $r$: there is no room for anything else.