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I was reading about the congruence modulo $n$ in The Integers from Topics in Algebra by I.N. Herstein; after showing the congruence modulo relation is equivalent, he introduced the set of all congruence classes viz. $J_n$ as:

[...] Let $J_n$ be the set of the congruence classes mod $n$; that is, $J_n = \{[0],[1],\ldots, [n-1]\}\:.$ Given two elements, $[i]$ and $[j]$ in $J_n,$ let us define \begin{align}[i]+[j] &= [i+j]\tag a\\ [i][j]&= [ij]\tag b\end{align} [...]

He then told they can be shown to be well-defined and jot down few properties which are left as exercise.

However, I didn't get how the author got motivated by "defining" whatever they are in that way above.

He didn't tell us why they are true or whether they are valid; the definition just came from nowhere.

Could anyone shed some light on how to show the relations are true? Or is it that they are defined so?

  • Maybe it would be clearer to say that there exists a ring homomorphism between $\Bbb Z$ and $J_n$ for every $n \in \Bbb N$ – cronos2 Nov 12 '16 at 10:08
  • I would appreciate if you elaborate @cronos2. –  Nov 12 '16 at 10:16
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    The objective is to endow $J_n$ with a ring structure. The difficulty is to make sure the definition is well defined. This means that no matter the value of $i+j$ (respectively $i\cdot j$) that is not necessarily $\lt n$ we get $[i+j]\in J_n$ (respectively $[i\cdot j]\in J_n$) and what we get is unique. It is obvious but requires to be noted. – marwalix Nov 12 '16 at 10:49
  • @marwalix, Herstein indeed shows the relations are well-defined in in the following discussion; but he didn't introduce the concept of ring then. I'd love to see your comment as an answer and would appreciate how it is related with the concept of ring. –  Nov 12 '16 at 10:55

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The objective is to endow $J_n$ with a ring structure. Indeed to have a ring you need two operations and you can check that the ring axioms are verified, the neutral for addition being $[0]$ and the neutral for multiplication being $[1]$.

The difficulty is to make sure the definition is well defined. This means that no matter the value of $i+j$ that is not necessarily $\lt n$ we get $[i+j]\in J_n$ and what we get is unique. It is obvious but requires to be noted.

marwalix
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  • Thanks for making the comment as an answer; +1 for that' I'll accept that when I start studying ring theory and have a sufficient rigor to handle problems on the same. –  Nov 13 '16 at 16:12