Ring $R_{ 118 }$

Description:

We take a free algebra $A$ over $\mathbb Q$ in noncommuting indeterminates indexed by $\mathbb N\times \mathbb N$. Define $\alpha(x_{i,j})=x_{i+1,j}$ and $\delta(x_{i,j})=x_{i,j+1}$. Let $D$ be the division ring of fractions of $A$. $\alpha$ and $\delta$ extend to $D$ and satisfy $\alpha\delta=\delta\alpha$. $\alpha$ is an endomorphism that is not an automorphism, and $\delta$ is a derivation that is not inner, and the ring required is the opposite ring of the twisted polynomial ring $D[t;\alpha,\delta]$ (where the product is defined $ct:=t\alpha(c)+\delta(c)$ for all $c\in D$.)

Keywords free algebra twisted (skew) polynomial ring