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I'm sorry if this is a dumb question, but I'm honestly confused. I was reading my modern algebra textbook and it started talking about the multiplication and addition table of $\mathbb Z/6\mathbb Z$. I understood the addition table, but the multiplication table had me confused. For example, why is the multiplication of $0+6\mathbb Z$ and $0+6\mathbb Z$ equal to $0+6\mathbb Z$? My initial impression was that it would equal $36\mathbb Z$ because $6*0=0$, $6*6=36$, $6*12=72$, etc. What aspect of my reasoning is faulty? Thank you in advance.

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  • $\begingroup$ By definition,$\Bbb Z/6\Bbb Z$ is the set of cosets $a+6\Bbb Z$, but $0+36\Bbb Z$ is not such a coset, so isn't an element of $\Bbb Z/6\Bbb Z$. $\endgroup$ – Lord Shark the Unknown Oct 29 '18 at 7:49
  • $\begingroup$ Thank you for your response. I recognize that $0+36\mathbb Z$ doesn't make sense to be an element of $\mathbb Z/6\mathbb Z$. I guess my main question is how you would perform multiplication of the elements of $\mathbb Z/6\mathbb Z$ because it seems counter-intuitive to me. $\endgroup$ – KronZ Oct 29 '18 at 7:58
  • $\begingroup$ My assumption is that in order to multiply $0+6\mathbb Z$ with $0+6\mathbb Z$, you need to multiply every element of these sets together and see what set comes out. But when I do this, I get $0+36\mathbb Z$ which doesn't make sense. $\endgroup$ – KronZ Oct 29 '18 at 8:08
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That is precisely the beauty of ideals, thez are stable under arbitrary multiplication. hence your cosets are also stable under multiplication. this is actually a really nice exercise to prove that a $R\backslash I$ is a ring if and only if $I\subset R$ is an Ideal (well definedness of multiplication here then actually boils down, to $r0=0$ for all $r \in R$. So in your concrete case that means that $$(2 + 6\mathbb{Z})*(2+6\mathbb{Z})= 4 + 6\mathbb{Z} + 6\mathbb{Z} +36\mathbb{Z}$$ but now since all the factors containing $\mathbb{Z}$ are contained in $6\mathbb{Z}$ this actually is $$4 + 6\mathbb{Z}.$$ Hence you can actually just multiply as in $\mathbb{Z}$ and then taking modulo $6$.

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