Determine the units in Z/12Z (confused about notation)

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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 $$U=\{[1],\;[5],\;[7],\;[11]\}.$$