# Ring detail

## Name: Semicommutative $R$ such that $R[x]$ is not semicommutative

Description: Let $A$ be the elements of the free algebra $F_2\langle a_0, a_1, a_2, b_0, b_1, b_2, c \rangle$ with constant term zero, and $F_2$ is the field of two elements. $A$ is a ring without identity. Let $I$ be the ideal of $F_2+A$ generated by $a_0b_0; a_1b_2 + a_2b_1; a_0b_1 + a_1b_0; a_0b_2 + a_1b_1 + a_2b_0; a_2b_2; a_0rb_0; a_2rb_2;(a_0 + a_1 +a_2)r(b_0 + b_1 + b_2)$ with $r \in A$ and $r_1r_2r_3r_4$ with $r_1,r_2,r_3,r_4 \in A$. The ring is $(F_2+A)/I$

Keywords free algebra quotient ring

Reference(s):

• C. Huh, Y. Lee, and A. Smoktunowicz. Armendariz rings and semicommutative rings. (2002) @ Example 2 p 753

Legend
• = has the property
• = does not have the property
• = information not in database

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