I need to find the list of all units in the ring $\mathbb Z_{12}$.
I know that there must be a multiplicative inverse, $a_{-1}$, for there to be units, $a$, and that
$$a\cdot a^{-1} = a^{-1} \cdot a = 1.$$
I think the multiplicative inverse here is $1$ and the answer tells me that the units are $\{1, 5, 7, 11\}$, but I am having trouble understanding.
Best Answer
The multiplicative identity is $1$, as (I think) you meant. Each number is allowed to have its own inverse, so we check. $1$ clearly divides itself, so $1$ is always a unit. $5 \cdot 5 = 25 = 1$, so we see that $5$ is a unit. $7 \cdot 7 = 49 = 1$, so $7$ is a unit. And $11 \cdot 11 = 121 = 1$, so it's also a unit.
You might have noticed a few things - we happen to have squared each of these numbers, so that they are each their own inverse. Also, the primes that divide $12$ are $2$ and $3$, and $5,7,11$ are exactly the numbers that are coprime to $12$. I don't know what you've learned so far, but neither of these are coincidences (really, it's a multiplicative group now too, so perhaps your group theory can kick in).