I want to determine the units of the Ring $\mathbb{Z}/12\mathbb{Z}$. But I am very confused about this notation. Can someone tell me what ring this is?
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See also I don't know if this could be helpful: http://math.stackexchange.com/questions/1253187, http://math.stackexchange.com/questions/1145015, http://math.stackexchange.com/questions/1826188 for the notation and intuition. – Watson Jan 14 '17 at 15:59
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This is the ring of integers modulo $12$, some times also written $\Bbb Z_{12}$. The reason it's written $\Bbb Z/12\Bbb Z$ is that $12\Bbb Z$ (i.e. the set of integers that are divisible by $12$) is an ideal in the ring $\Bbb Z$ of integers, and ideals let you make quotient rings the same way normal subgroups let you make quotient groups.
Arthur
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So it's asking to find elements in the ring such that if we multiply them together, we get 1? – Lana Jan 14 '17 at 15:24
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@Lana More specifically, the question is asking you to find those elements in the ring for which there exists such a "multiplies-to-1" partner (which may, but doesn't have to be, be itself). You don't have to explicitly find the full pairs, although that is a valid way of doing it. I can begin by saying that $11$ is a unit, because $11\cdot 11 = 1$. Now you just have to find the rest. – Arthur Jan 14 '17 at 15:25
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Have a look at Quotient ring. An invertible element or a unit in a unital ring $R$ is any element that has an inverse element with respect the multiplication. In our case. $$\mathbb{Z}/{12\mathbb{Z}}=\{[0],\;[1],\;[2],\;\ldots,\;[11]\}$$ For example $[1]\cdot [1]=[1]\Rightarrow [1]\text{ is unit}.$ We can verify that the set of all units of $\mathbb{Z}/{12\mathbb{Z}}$ is $$U=\{[1],\;[5],\;[7],\;[11]\}.$$
Fernando Revilla
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This is quotient group, where $12Z$ stands for $\{12\cdot z : \; z\in \mathbb{Z} \}$. You can think of these elements as of elements of $\{0,1,2,3,4,5,6,7,8,9,10,11 \}$, since there is a natural isomorphism between those two rings.
