In a ring $R$ for every natural number $n$, $I_n$={$na:a$ $\in$ $R$} is defined.
prove that $char(R/I_n)$ divides $n$ where $char$ denotes the characteristic of a ring.
It is easily shown that $I_n$ is an ideal and also I know the definition of $char$ but I can not prove this.
It is a problem from book $Modern$ $Algebra$ by $Surjeet$ $Singh$ and $Qazi$ $Zameeruddin$.
Thank for your time.
Best Answer
Thankful to @CaptainLama
lemma: For an arbitrary ring $A$ with $\operatorname{char}(A)=m$ such that $m\in\mathbb{N}$ if $na=0$ for all $a\in A$ then
$\operatorname{char}(A)|n$.
proof: From the definition of characteristic we have $n\ge m$. For $n=m$ it is obvious, so suppose that $n\gt m$ and also $\operatorname{char}(A)\not|n$, now from division algorithm $n=qm+r$ for $q\in\mathbb{Z}$ and $0\lt r\lt m$. Substituting for $n$ implies that $ra=0$ for all $a\in A$ and a contradiction; because $m$ was the smallest natural number with this property.
From the lemma, It suffices to show that $n(r+I_n)=I_n$ for all $r\in R$.
$n(r+I_n)=nr+I_n$ from definition of $I_n$ we have $nr\in I_n$ finally $n(r+I_n)=I_n$ for all $r\in R$ and $\operatorname{char}(R/I_n)|n$.