Is there name for Ring, with the following Cayley tables I'm solving a Cayley table for Ring with four elements and noticed that the resulting multiplication table looks pretty interesting. It has two left multiplication 1s and two left multiplications zeros and one of those zeros is ofcourse the zero in the Additive group of the Ring. Is there any special name for this Ring?
Let R = {a, b, c, d}
$$
\begin{array}{ l | c c c c  }
   + & a & b & c & d \\ \hline
   a & d & c & b & a \\
   b & c & d & a & b \\
   c & b & a & d & c \\
   d & a & b & c & d
\end{array} \quad \quad \quad \quad
\begin{array}{ l | c c c c  }
   * & a & b & c & d \\ \hline
   a & ~ & d & d & d \\
   b & ~ & ~ & ~ & d \\
   c & a & b & ~ & d \\
   d & d & d & d & d
\end{array} \\
cc = c(a + b) = ca + cb = a + b = c \\
aa = a(c + b) = ac + ab = d + d = d \\
ba = (c + a)a = ca + aa = a + d = a \\
bb = (c + a)b = cb + ab = b + d = b \\
bc = (c + a)c = cc + ac = c + d = c \\
\begin{array}{ l | c c c c  }
   + & a & b & c & d \\ \hline
   a & d & c & b & a \\
   b & c & d & a & b \\
   c & b & a & d & c \\
   d & a & b & c & d
\end{array} \quad \quad \quad
\begin{array}{ l | c c c c  }
   * & a & b & c & d \\ \hline
   a & d & d & d & d \\
   b & a & b & c  & d \\
   c & a & b & c & d \\
   d & d & d & d & d
\end{array}
$$

I've never heard a name for this rng, but I have seen it expressed this way:
Given the two-element semigroup $S=\langle b,c\mid b^2=cb=b, c^2=bc=c\rangle$, your rng is the semigroup rng $F_2[S]$.
