Which of the following is NOT a proper ideal group of $ \mathbb{Z}_{12} $ ?
a)<5>
b)<8>
c)<2>
d)<3>
e)<4>
f)none
Currently reviewing abstract algebra and I am not sure how to go about this question. Can someone explain what an ideal group is in detail?
I have no idea what an ideal group is, but I know what an ideal is. $(a)= \mathbb{Z}_n$ if and only if gcd$(a,n)=1$. This is because of the Bezout property of the integers. In this case, there are integers $s,t \in \mathbb{Z}$ with $as+nt=1$ so $as \equiv 1$ in $\mathbb{Z}_n$ and is a unit. Any ideal containing a unit in a ring will necessarily be the entire ring since if $\lambda \in U(R)$, then $r=(r\lambda^{-1})\lambda$. So if you pick one of those which is relatively prime to $12$, it will generate the whole ring and not be a proper ideal.