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What is the difference between ring $\mathbb{Z}_n$ and congruence to modulo $n$.

Are they same thing or what are significant differences between them except we can use integers greater than $n$ for modulo operations. They look like the same.

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    Congruence modulo $n$ is a _relation_, so it cannot be a _ring_. They are entirely different types of things; this alone should suffice as an answer.2012-11-02
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    @Marc That's not true generally - see my comment to your answer.2012-11-02
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    @BillDubuque: just for clarity: by "That" you apparently mean "Congruence is a relation". You claim congruence _can_ be defined as a subalgebra (of $\Bbb Z\times\Bbb Z$ in this case). That's fine with me, although it's not likely that OP considers it this way. And even then, this subalgebra is rather different from $\Bbb Z_n$.2012-11-02
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    @Marc You stated that "a congruence cannot be a *ring*". That is false. In fact that a ring congruence *is a ring* is an *essential* property of a congruence. One cannot argue such semantic points so simply.2012-11-02
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    @BillDubuque: As I said, it's fine with me; have a nice day. But I actually meant to say a relation cannot be a ring; maybe that is not even quite true set-theoretically, but I don't care.2012-11-02

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