I just started learning about rings, and when playing around with some examples I noticed that in all of them, every subgroup of their additive group was also a subring.
So, I'm looking for examples where that is not the case (a subset of a ring being closed under addition and negatives but not under multiplication).
I tried looking for some but only found a question asking for subgroups of the additive group that aren't ideals.
Best Answer
Consider the group of $2\times 2$ real matrices, if you consider the additive subgroup generated by the matrix $$ \begin{bmatrix} 1 &1\\ 0 & 1 \end{bmatrix} $$ The additive group it generates is isomorphic to the integers, but it is not a sub ring since if you square the matrix you get $$ \begin{bmatrix} 1&2\\ 0&1 \end{bmatrix} $$ which is not in the additive group it generates.