Generic matrix algebra

Summary

The generic matrix algebra is the algebra generated by finitely many $n \times n$ matrices whose entries are independent commuting variables. It is the "most general" algebra generated by $m$ matrices of size $n$. For every commutative $k$-algebra $C$, the $k$-algebra generated by $m$ matrices in $M_n(C)$ is a quotient of it. The quotient map substitutes elements of $C$ for the variables. Over an infinite field $k$, it is the relatively free algebra of the variety of algebras that satisfy all identities of $M_n(k)$. It is a domain. It is a basic object in the theory of polynomial identity rings and in the invariant theory of matrices.