What does it mean to say R[A,B] where R is a ring and A, B are square matrices with entries in R?

Question

$R[A]$ makes sense when $A$ is a square matrix but what is $R[A,B]$?

Answer 1

It is the ring of $R$-linear combinations of arbitrary products of $A$ and $B$. This will contain elements like $A^2B-B^2A$ and $B^2A^3BA+AB+B^2$. This is not a quotient of the usual polynomial ring in two variables, but it is a quotient of a polynomial ring in two noncommuting variables.