# Elements of the ring $\mathbb Z_{15}$

There is something I don’t understand when I want to write the elements of the ring $\mathbb Z_{15}$.

Consider the ring $\mathbb Z_{15}$ with a unity $1$. Show that $5\mathbb Z_{15}$ is a ring with a unity $10$.

Will I be right to say $5\mathbb Z_{15}=\{0,5,10\}$?

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What does "again" refer to? – joriki Jan 29 '12 at 7:50
I don't understand how does one define the ring $5\mathbb{Z}_{15}$? – user38268 Jan 29 '12 at 7:52
@BenjaminLim: I presume $5\mathbb Z_{15}=\{5a:a\in\mathbb Z_{15}\}$. – Jonas Meyer Jan 29 '12 at 7:56
@JonasMeyer Ok thanks. – user38268 Jan 29 '12 at 8:04

Yes, you are right in saying that $5\mathbb{Z}_{15}=\{0,5,10\}$ since this is the set obtained by multiplying each element in $\mathbb{Z}_{15}$ with $5$. Now you need to check that this set is a ring with the operations inherited from $\mathbb{Z}_{15}$. Note that you need check only closure and the existance of the additive and multiplicative identities. This is because if the other axioms failed in this ring, they would fail in the larger ring containing it, which is impossible since we know that $\mathbb{Z}_{15}$ is a ring.