How many proper non-zero ideals does the ring $\mathbf{Z}_{12}$ have? How many ideals does the ring $\mathbf{Z}_{12}\bigoplus\mathbf{Z}_{12}$ have?
I think that $\mathbf{Z}_{12}$ should have $4$ proper ideals given by following $$\langle 2 \rangle=\langle10\rangle,\langle3\rangle=\langle9\rangle,\langle4\rangle=\langle8\rangle,\langle6\rangle$$
But I am not sure how to do second part as I cant assume that every ideal will be of the form $\langle(a,b)\rangle$ for some $a,b\in\mathbf{Z}_{12}$. Because it would mean that the direct product ring is a principal ideal ring.
Any help is appreciated. Thanks
Best Answer
The ideals of $\mathbf{Z}_{12}$ are in bijection with the ideals of $\mathbf{Z}$ containing $12\mathbf{Z}$. Since the divisors of $12$ are $1$, $2$, $3$, $4$, $6$ and $12$, there are six of them. Now the answer depends on the meaning you assign to “proper”.
The ideals in $R\oplus S$ are of the form $I\oplus J$, where $I$ and $J$ are ideals of $R$ and $S$ (proof?).