Ring $R_{ 133 }$

Description:

Let $X$ be an infinite set and let $M$ be the monoid with zero given by the presentation $$\langle X\cup X^*\mid x^*y=\delta_{x,y}, x,y\in X\rangle$$ where $X^*$ is a bijective copy of $X$. Let $R=K_0M$ be the contracted monoid algebra. So it has $K$-basis $M\setminus \{0\}$ and the product extends that of $M$, where we identify the zero of $M$ with the zero of $K$. Now define $d\colon R\to R$ on the basis $M\setminus \{0\}$ as follows. Fix $x\in X$ and put $$d_x(pq^*) = (|p|_x-|q|_x)pq^*.$$ Here $|w|_x$ is the number of occurrences of the letter $x$ in $w$. $d$ is a noninner derivation.

Keywords Contracted monoid ring Leavitt path algebra