abstract algebra - Requirements on elements for the inversion of matrices to be well defined.

I'm trying to learn some algebra here. Taken a few more or less clumsy steps on groups and in some fields.

If I have a matrix with elements from some set, what are the requirements on the set and the operations addition and multiplication to guarantee that matrix inverse exists in some sense?

Own work Well at least some thoughts I have had...

1. Addition and multiplication must be well defined on the elements, or how else would we define matrix multiplication in the first place? It is usually defined recursively on the elements as sums of products, right..?


2. Additive and multiplicative identity elements should be necessary - or what else should "0" and "1" in the $I_n$ mean?


3. Likewise I guess additive and multiplicative inverses should exist, but I lack some arguments for it.


4. Is associativity necessary? Also here I lack arguments for or against.