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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $m, n \\ge 1$ be integers.\nLet $P = k\\big[x^{(l)}_{ij} : 1 \\le l \\le m, \\ 1 \\le i, j \\le n\\big]$ be the [[polynomial ring|0SDC934R52]] in $m n^2$ commuting variables.\nFor $1 \\le l \\le m$, the *generic matrix* $X_l \\in M_n(P)$ is the [[matrix|0JXZKX526R]] whose $(i, j)$ entry is $x^{(l)}_{ij}$.\nThe *generic matrix algebra* $k\\{X_1, \\ldots, X_m\\}$ is the $k$-subalgebra of $M_n(P)$ generated by $X_1, \\ldots, X_m$ and the identity matrix.\n\n## Remarks\n\nThe algebra $k\\{X_1, \\ldots, X_m\\}$ is an [[associative algebra|3CTW1P4HBP]] over $k$, and it is a subring of $M_n(P)$, so it satisfies every [[polynomial identity|C0EDT81EGH]] of $M_n(P)$.\nLet $\\pi \\colon k\\langle x_1, \\ldots, x_m \\rangle \\to k\\{X_1, \\ldots, X_m\\}$ be the homomorphism from the [[free algebra|2P3Y2FRCSZ]] with $\\pi(x_l) = X_l$.\nIf $k$ is infinite, the kernel of $\\pi$ is the set of polynomial identities of $M_n(k)$ in $m$ variables.\nIndeed, $f(X_1, \\ldots, X_m) = 0$ means that each entry of the matrix $f(X_1, \\ldots, X_m)$ is the zero polynomial.\nOver an infinite field, a polynomial is zero exactly when it vanishes at all points, that is, when $f(a_1, \\ldots, a_m) = 0$ for all $a_l \\in M_n(k)$.\n\nExample: for $n = 1$ the generic matrices are the variables $x^{(l)}_{11}$, and $k\\{X_1, \\ldots, X_m\\}$ is the polynomial ring in $m$ variables.\n\nNon-example: for $n \\ge 2$, the scalar matrix $x^{(1)}_{11} I$ lies in $M_n(P)$ but not in $k\\{X_1, \\ldots, X_m\\}$.\nThe entries of $f(X_1, \\ldots, X_m)$ have degree-$1$ part equal to the corresponding entry of $\\sum_l c_l X_l$, where $c_l$ is the coefficient of $x_l$ in $f$.\nIf $f(X_1, \\ldots, X_m) = x^{(1)}_{11} I$, comparing the $(1, 1)$ entries gives $c_1 = 1$, and comparing the $(2, 2)$ entries gives $\\sum_l c_l x^{(l)}_{22} = x^{(1)}_{11}$, which is impossible.\nSo the subalgebra generated by the $X_l$ over the commutative ring $P$ is strictly larger than the generic matrix algebra.\n\nFor $n \\ge 2$ and $m \\ge 2$, the generic matrix algebra is a noncommutative domain.\n",
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