Nilpotent ring example

Question

I have recently learned about nilpotency in a group/ring.I am thinking about doing some work on a non-commutative nilpotent ring but i could not find an example. More precisely, (S,+,.) is nilpotent, where S is nilpotent w.r.t. both of it's operation.

Answer 1

Take a set of matrices $n\times n$, $n>3$ such that $\{a_{1n-2},a_{1n-1},a_1n,a_{2n-1},a_{2n},a_{3n}\}\subset\mathbb R$ and $a_{ij}=0$ for any $(i,j)\not\in\{(1,n-2),(1,n-1),(1,n),(2,n-1),(2,n),(3,n)\}.$

Answer 2

Start with a field of $3$ elements $F_3$ and take the free algebra $F_3\langle x,y\rangle$ in noncommuting indeterminates, then take the quotient by $(x,y)^3$, and let $R$ be the ideal $(x,y)/(x,y)^3$ in the quotient ring.

It's nilpotent because $a^3=0$ for everything in the ring, and it's not commutative because $xy\neq yx$.

It also satisfies $3a=0$ for every $a\in R$, if that is what you meant by "additively nilpotent" (it should actually just be said that everything has additive order less than $3$.)