If $I$ is an ideal of a ring $R$, then for any $a\in R$, is there a difference between $(R/I)a$ and $Ra/I$?

Question

If $I$ is an ideal of a ring $R$, then for any $a\in R$, is there a difference between $(R/I)a$ and $Ra/I$?

I am seeing the following:

$$(R/I)a=\{r+I: r\in R\}a=\{ra+I:r\in R\}=Ra/I.$$ Does this make sense?

Answer 1

Mostly, but not quite. There is an issue that maybe $Ra$ doesn't contain $I$, so the right hand side needs more careful definition.

You could write that $(R/I)a=(Ra+I)/I$, though.

For a concrete example, $R=\mathbb Z$, $a=2$ and $I=3\mathbb Z$. In that case $(R/I)a=R/I$, but you wouldn't write $2\mathbb Z /3\mathbb Z$.