notation in congruence relation

hi there i was looking through my lecture notes and i'm struggling to understand a particular piece of notation the vertical line | and i was wondering if you could explain its meaning

$$f \sim g \iff f - g \text{ is an element of } (x^2) \iff x^2|f-g$$

where $f$ and $g$ are elements of polynomial ring $R[x]$ and $(x^2)$ is an ideal s.t. $\{f\cdot x^2 \text{ is an element of } R[x] \mid f \text{ is an element of } R[x]\}$

see i understand the second use of | but not the first. could anyone explain the meaning to me please

• What is the first use of $|$? That $x^2\mid f(x)-g(x)$? This means that $x^2$ divides $f(x)-g(x)$. – Dietrich Burde Mar 2 '18 at 19:15
• The first means "divides evenly". – lulu Mar 2 '18 at 19:15
• @DietrichBurde Oh, I just meant "divides". In English we often say things like $3$ divides $9$ evenly, to distinguish it from situations like $\frac 83\in \mathbb Q$. – lulu Mar 2 '18 at 19:18
• I see, "a divides b evenly", e.g., here. I was confused by the word "even" (to divide oddly) – Dietrich Burde Mar 2 '18 at 19:18
• @DietrichBurde exactly. Here "evenly" has nothing to do with parity...it's "even" in the sense of uniform or fair. As in, "you have a bunch of toys and you wish to distribute them to the kids evenly." – lulu Mar 2 '18 at 19:19

The first use means 'divides' — thus, there is a polynomial $h(x)$ such that $f(x)-g(x) = x^2h(x)$.
Also, to ensure proper spacing use the \mid command; e.g., $x^2 \mid f(x) - g(x)$.
Finally, since $x$ is being used as a variable, it's a good idea to write $f(x)$ instead of $f$ in this context since the polynomial $x^2$ doesn't have a name.