2012-31.md

course	course_year	question_number	tags	title	year
Groups, Rings and Modules	IB	31	IB 2012 Groups, Rings and Modules	Paper 4, Section I, $2 G$	2012

An idempotent element of a ring $R$ is an element $e$ satisfying $e^{2}=e$. A nilpotent element is an element e satisfying $e^{N}=0$ for some $N \geqslant 0$.

Let $r \in R$ be non-zero. In the ring $R[X]$, can the polynomial $1+r X$ be (i) an idempotent, (ii) a nilpotent? Can $1+r X$ satisfy the equation $(1+r X)^{3}=(1+r X)$ ? Justify your answers.