Conditions for a certain ideal in a ring of functions $R^X$ to be principal

Question

Assume that $R$ is a commutative ring with a multiplicative identity. Let $X$ be a set, and let $Y\subset{X}$. The set ${R^{X}}$ which consists of all functions $X\rightarrow{R}$ is a ring under the binary operations $(f+g)(x)=f(x)+g(x),$ and $(fg)(x)=f(x)g(x)$. Let $I=\{f\in{R^{X}}:f(Y)=0\}$. Under what conditions on $Y$ is $I$ a principal ideal in $R^{X}$? One condition I found was $Y=X$. In this case $I=(0)$. I can't think of other conditions on $Y$ in this general setup that would enable us to conclude that $I$ is a principal ideal.

Answer 1

The functions $X \to R$ that vanish on a fixed subset $Y$ of $X$ always form a principal ideal for any set $X$ and any ring $R$. In fact, it is the principal ideal generated by the indicator function of the complement of $Y$.