Congruence mod n is a congruence relation for integer addition. Is the converse true? That is, is every congruence relation for integer addition of the form mod n for some integer n?
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What exactly do you mean by congruence relation? One well defined under the addition of $mathbb Z$? – paul blart math cop Mar 24 '20 at 18:34
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@paulblartmathcop An equivalence relation R such that aRb and cRd implies a+cRb+d – user107952 Mar 24 '20 at 18:37
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Generally for any commutative ring $R$, a quotient by a congruence corresponds to the quotient ring by the ideal of all elements $\equiv 0$. Further a congruence is the same as a sub-$R$-algebra of the square $R^2$. This is explained at length in this answer in the linked dupe. – Bill Dubuque Mar 24 '20 at 18:55
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The algebraic structures (likes rings and groups) whose congruences are determined by by a single congruence class are known as ideal determined varieties. – Bill Dubuque Mar 24 '20 at 19:01
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Of course the trivial relation, $a\sim b\iff a=b$ is a congruence relation in this sense.
Let's assume that there is some pair, $a>b $, with $a\sim b$. Then $(a-b)\sim 0$ so there is some positive integer $n$ such that $n \sim 0$. Let $n$ be the least positive integer with $n\sim 0$.
We then contend that $a\sim b \iff n\,|\,(a-b)$.
Note first that $\Leftarrow$ is clear.
Now suppose $a\sim b$ with $a>b$. We write $(a-b)=nq+r$ where $0≤r<n$. It is easy to see that $r\sim 0$ so if $r\neq 0$ we'd contradict the minimality of $n$ and we are done.
lulu
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