Let $R$ be a group/ring and $I$ a normal subgroup/ideal, and form the quotient group/ring $R/I$. Is is legitimate to write either of the following?
$S/I$ is an arbitrary subset of $R/I$, where $S \subseteq R$.
$S/I$ is an arbitrary subset of $R/I$, where $I \subseteq S \subseteq R$.
If not, is there a way to represent an arbitrary subset (or subgroup or subring) of $R/I$ without just calling it $A$ or some other single variable? My question stems from wanting to find a way to represent a subset of the quotient ring before I actually check that it's an ideal.
Some people like to use that notation, and some people hate it. I personally think the best way to go is to take your arbitrary subset $A \subseteq R/I$ and define $S = \pi^{-1}(A)$ where $\pi : R \to R/I$ is the canonical projection. Then $S/I$ is guaranteed to be $A$ and you don't have to worry about people disliking your notation.