Ring $R_{ 111 }$

Left-not-right uniserial domain


Let $k$ be a field, and $F$ be the field of rational functions for $k(t_0, t_1,\ldots)$ in countably many variables. $\alpha$ is the injective homomorphism such that $\alpha(t_i)=t_{i+1}$, and $t_0$ is transcendental over $\alpha(F)$. Let $A$ be the ring generated by $\frac{f}{1+gy}$ where $f,g\in A[y]$. We can extend $\alpha$ to $\phi:A\to A$ by the rule $\phi(y)=t$. The ring is $R=A[[x;\phi]]$, the skew-powerseries ring using $xa=\phi(a)x$.

Keywords subring twisted (skew) polynomial ring


  • A. A. Tuganbaev. Semidistributive modules and rings. (2012) @ Example 9.10 p 213

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Name Measure
global dimension left: 1right:
Name Description
Idempotents $\{0,1\}$
Left singular ideal $\{0\}$
Left socle $\{0\}$
Nilpotents $\{0\}$
Right singular ideal $\{0\}$
Right socle $\{0\}$
Zero divisors $\{0\}$