Given $x\in R$, $R$ a commutative ring, what is the difference between $R/(x)$ and $coker(R\rightarrow^{*x} R)$ and how do I visualize the latter?

Question

Exactly what it says in the title. Here $\rightarrow^{*x}$ denotes just the endomorphism given by multiplication with x. Of course the two live in different categories, but are the underlying sets different? As far as I understand it, both are just the quotients given by the relation $x=0$.

Answer 1

Let $\mathsf{Ring}$ be the category of unital commuative rings. The cokernel of any morphism in $\mathsf{Ring}$ is the zero ring.

Though, if you consider $R$ as an $R$-module, and work inside the category $\mathsf{Mod}_R$, then $$R\stackrel{f}\longrightarrow R,\quad f(r)=rx$$ has cokernel $R/\operatorname{im}(f)=R/Rx=R/(x)$.