Question on notation from Eisenbud's Commutative algebra - (x:y)

Question

Just a simple question. What does Eisenbud mean by $(x:y)$ where $x,y \in R$ a ring. An example on this is in the section 17 discussing the homology of the koszul complex. I assume it's something along the lines of $\{r \mid ax = ry \}$ for some $a \in R$.

Answer 1

Eisenbud introduces this notation in section 0.3. If $I$ and $J$ are two ideals in $R$, then $$(I:J) = \{ r\in R\mid r\cdot J \subseteq I \}.$$ In this case, $I = (x)$, $J = (y)$.