Where did the notation $\Bbb Z/n\Bbb Z$ came from? By this I mean the ring $(\Bbb Z, +_{\bmod n},\cdot _{\bmod n})$.
Shouldn't the "$n\Bbb Z$" part be an equivalence relation(to quotient the set?)?
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1the / means quotient group, nZ is just multiplying every element of Z by n point-wise, giving a new group – Asier Calbet Sep 24 '15 at 17:41
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3$n\mathbb{Z}$ denotes the ideal (https://en.wikipedia.org/wiki/Ideal_(ring_theory) ) of multiples of $n$, and the slash denotes quotienting by this ideal. – Qiaochu Yuan Sep 24 '15 at 17:42
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The equivalence classes are the cosets, which are sets of the form $m + n \Bbb Z$ for $m \in \Bbb Z$. – Ben Grossmann Sep 24 '15 at 17:45
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It is interesting to read about the history of quotient groups $G/N$, see here. – Dietrich Burde Sep 24 '15 at 17:50
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I think it's just part of a more general notation for cosets of a subgroup, which has been popular for most of the 20th century. If $G$ is a group, and $H$ is a normal subgroup, then $G/H$ is the set of cosets $xH : x \in G$.
With this notation, an integer modulo $n$ is literally a subset of $\mathbb{Z}$. For example $1$ modulo $\mathbb{Z}$ is the subset $$\{ ... , 1, n+1, 2n + 1, ...\} = 1 + \{... , 0, n, 2n, ...\} = 1 + n \mathbb{Z}$$ and so the integers modulo $n$ are just $$\{ k + \{... , 0, n, 2n, ...\} : k \in \mathbb{Z}\} = \{ k + n \mathbb{Z} : k \in \mathbb{Z} \} = \mathbb{Z}/n\mathbb{Z}$$
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