How should one interpret $R[A_1,..,A_n]$ where $R$ is a unital ring and $A_i\in M_{n\times n}(R)$?
Intuitively, I considered $R[A_1,..,A_n]$ as almost a set of polynomials where '$X_i$' is each '$A_i$' and the coefficient comes from $R$. (So almost like a multivariable polynomial)
However, the definition I was given was "the unital subring that is generated by scalar multiples of the identity matrix and $A_1$,...,$A_n$." This almost coincides with what I initially thought, except, the bold text is what confuses me.
I would just think it is "the unital subring that is generated by the identity matrix and $A_1$,...,$A_n$." Is the 'scalar multiples' part of the definition necessary? If so why is it necessary?