Meaning of the slash "/" in $\mathbb{Z}/p\mathbb{Z}$

If $R$ is a ring and $J$ a two-sided ideal of $R$, the quotient ring $R/J$ consists of the equivalence classes $x + J$ for $x \in R$, where $x \sim y$ if $x - y \in J$. This is a ring with operations $(x+J) + (y+J) = (x+y)+J$ and $(x+J)(y+J) = xy + J$.

In the case of $\mathbb Z / p\mathbb Z$, $p\mathbb Z$ consists of the multiples of $p$ and the equivalence relation is congruence mod $p$. Thus $\mathbb Z/p\mathbb Z$ consists of the congruence classes mod $p$.

In $\Bbb Z$ ,we define the equivalence relation $R$ by $x \;R\; y \iff x-y$ is a multiple of $p $.

$\Bbb Z/p \Bbb Z $ is the set of equivalence classes.

$$\Bbb Z/3\Bbb Z=\{\overline {0},\overline {1},\overline {2}\}. $$

$$\overline {3}=\overline {0}. $$

Risking self-praise, I might recommend my own course notes (formerly a book published by a traditional publisher) "Coding Notes" at http://www.math.umn.edu/~garrett/coding/CodingNotes.pdf There are also the rather telegraphic overheads for a course I taught many times on that subject using those notes/book, at http://www.math.umn.edu/~garrett/coding/

This course (and the book/notes) was meant to be intelligible to people who'd not studied any abstract algebra before, and, in particular, to engineering and computer science people, in addition to math majors in the relatively early part of their undergrad education.

So, in particular, these notes are very down-to-earth, and talk in a way precisely meant to be intelligible to engineering and computer science people... who may have a "different dialect" in mathematics.

So, no, not abstract, yet mathematically accurate, and aimed at coding-theory issues. (Though not really high-end, and certainly no longer up-to-date.)