While checking many sites about subrings and how to prove a subset of a ring $R$ is also a subring of it, I saw many sites asking if a set is a subring of a ring but even the operation in these two sets are different, and this is contradictory to the definition of a subring!
For example, I saw a question that asks if $\mathbb{Z_6}$ is a subring of $\mathbb{Z_{12}}$, and justifies that's not by showing that it's not closed under addition $mod$ $12$: $5+5= 10$ $mod$ $12$, but $10$ is not in $\mathbb{Z_6}$. I can't understand this way of disproving anymore, since $10$ in $\mathbb{Z_{12}}$ is different from $10$ in $\mathbb{Z_6}$ which's $4$ $mod$ $6$, we are talking about different operations!! ( addition $mod$ $12$ and addition $mod$ $6$ ), so this is clear that $\mathbb{Z_6}$ is not a subring of $\mathbb{Z_{12}}$ without the need of a proof! So why are many sites dealing the same way with such questions?!
I have another one:
Can we consider $\mathbb{Z_p}$ a $subset$ of $\mathbb{Z_q}$ if $p\leq q$??
I thought this is right because while deciding whether a set is a subset of another set, we don't look at their operations, but I was shocked that one site writes: $\mathbb{Z_{2}}$ is $not$ $a$ $subset$ of $\mathbb{Z_{6}}$!
I feel like I have many new problems in abstract algebra, please help me to reorder my info !