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The ring operations are exactly the same and the elements are exactly the same, so does the slightly different way they are constructed matter?

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What is that "slightly different" way they're constructed? One is just a short form of writing the other one. – DonAntonio Feb 5 '13 at 3:39
What does the congruence relation mean? It means to take $n$ to be $0$, i.e. to form the quotient ring. So the two constructions coincide in fact. – awllower Feb 5 '13 at 3:43
Well I guess what I want to ask then is are they the same because there is an obvious isomorphism or are they literally the same thing? – user61031 Feb 5 '13 at 3:45
"The elements are the same and the operations are the same" is the definition of ring isomorphism... – vonbrand Feb 5 '13 at 4:14

Both $\mathbb{Z}_n$ and $\mathbb{Z}/n\mathbb{Z}$ should be interpreted as meaning exactly the same thing. The distinction is only at the level of notation.

They are not just isomorphic rings, but are, in fact, equal.

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