A set, two binary operations, but multiple identities is called a ...? 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.
 A: Of course this all depends on the operations which you have been a little sketchy about:
(from the comments)

By way of ⊗ and ⊕, which are defined to extend the sizes of the matrices so that addition and multiplication is possible in the usual way.

That could be done in many ways.
If you are, for example, just padding the smaller square matrix with zeros and then adding/multiplying as normal, then I think you are just looking at the direct limit of these rings in the category of rings without identity.
If you padded with zeros except for $1$'s along the diagonal, I think that makes it a direct limit in the category of rings with unity.
(I've heard of another construction that takes the LCM of the sizes of the matrices, and then uses block-diagonal matrices build from the original two so that you have two matrices with the LCM as the size which you can multiply and add. This seems more complicated than you probably intended.)
The thing is that with all these approaches, not all of the things you included ("all of the $n\times n$ identity matrices"/"all of the $n\times n$ zero matrices") stay distinct. If one matrix "extends" to a second, then those two things should be considered equal. So there is still only one additive identity, the common limit of all the finite additive identities. If you padded with $1$'s on the diagonal, there is still only one multiplicative identity, the common limit of all the finite identities.
