First of all, when I say that $(R, +, \cdot)$ is a ring, I mean $R$ commutative ring with identity and I am using the definition that $(S, +, \cdot)$ is a subring of $R$ if $(S, +)$ is a subgroup of $(R, +)$ and S is closed under $\cdot$ and has an identity but that identity need not be the same as the idenity of $R$.
I was able to show that of if $\mathbb{Z}_{d}$ is a subring then $d\mid m$ and I expected the converse to be true as well (if $d \mid m$ then $\mathbb{Z}_{d}$ is a subring of $\mathbb{Z}_{m}$) but that fails if we take $m = 12$ and $d = 6$.
Is there a way to determine if $\mathbb{Z}_{d}$ is a subring of $\mathbb{Z}_{m}$ by just looking at the values of $d$ and $m$?
