# Ring detail

## Name: $k[x;\sigma]/(x^2)$ (Artinian)

Description: Let $\sigma:k\to k$ be a field endomorphism of a countable field $k$ such that $\infty > n=[k:\sigma(k)]>1$. $k[x;\sigma]$ is the twisted polynomial ring where $xa:=\sigma(a)x$ for all $a$ in $k$. The ring is $k[x;\sigma]/(x^2)$.

Notes: Composition length finite but different on both sides

Keywords quotient ring twisted (skew) ring

Reference(s):

• E. A. Rutter and Jr. Rings with the principal extension property. (1975) @ Example 1, pp 208-209
• W. K. Nicholson and M. F. Yousif. Annihilators and the CS-condition. (1998) @ Example 3.8 p 221

Legend
• = has the property
• = does not have the property
• = information not in database
Name Measure
cardinality $\aleph_0$
composition length left: 1right: $n$
Krull dimension (classical) 0
Name Description
Idempotents $\{0,1\}$