I have a set $R$ of $n\times n$ matrices where $n=1,2,3,\ldots$, and two binary operations $\oplus$ and $\otimes$. I can prove that $(R,\oplus,\otimes)$ is a "ring" if I am permitted to allow all the additive inverses $J_n$, the set of all $n\times n$ zero matrices, and allow all the multiplicative inverses $I_n$, the set of all $n\times n$ identity matrices (1's on the main diagonal, zeros elsewhere), i.e. all $I_n$ and $J_n$ where $n=1,2,3,\ldots$.
Can I actually claim this to be a ring, or does this fall outside the definition of ring. If so, what object is it?
I'm pretty sure it's not exactly a ring because the additive and multiplicative identities must be unique.
Update1 I think this could be related to the idea of multi-ring.
Update2 Perhaps a better reference of multiring is to be found here. Apparently, they were introuced in 1956, 1983 and then independently in 2006 (see [10,11,15] in reference list in ibid.) I think they're called hyperring and multiring synonymously, but not totally sure on that just yet.