For the curious, it’s not a “group” because the integers don’t have multiplicative inverses. If I have x=2, there is no integer that I can multiply that by to get 1. Integers with addition on the other hand is a group, which is a monoid with the additional property that inverses are present.

Additionally, if you restrict to invertible matrices with matrix multiplication, they are also a monoid (so they are a submonoid), and in fact a group, the general linear group. If you restrict again to matrices with determinant 1, it's again itself a group (a subgroup, and also a submonoid), the special linear group.

The operation is important too. Square matrices over integers with matrix multiplication is (just) a monoid. Square matrices with addition are a monoid too (but also a group, because there is an additive inverse).

Put it all together and it’s called a ring

Yes