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  "body": "\n## Statement\n\nLet $K$ be a [[commutative ring|0310REJJY8]] with identity and let $n \\ge 1$.\nFor $m \\ge 1$, the *standard polynomial* of degree $m$ is\n$$s_m(x_1, \\ldots, x_m) = \\sum_{\\sigma \\in S_m} \\sgn(\\sigma) \\, x_{\\sigma(1)} x_{\\sigma(2)} \\cdots x_{\\sigma(m)},$$\nwhere $S_m$ is the [[symmetric group|0515A6D2Y1]] and $\\sgn$ is the [[sign of a permutation|02DKV69ZH0]].\n\n1. For all [[matrices|0JXZKX526R]] $A_1, \\ldots, A_{2n} \\in M_n(K)$, one has $s_{2n}(A_1, \\ldots, A_{2n}) = 0$.\n2. If $K \\ne 0$, then $M_n(K)$ satisfies no monic [[polynomial identity|C0EDT81EGH]] of degree less than $2n$.\n\n## Remarks\n\nPart 2 is the easy part.\nA monic identity of degree $d \u003c 2n$ can be replaced by a multilinear monic one of the same degree.\nOne evaluates it on the first $d$ matrix units of the staircase $e_{11}, e_{12}, e_{22}, e_{23}, \\ldots, e_{n-1,n}, e_{nn}$.\nOnly the product in the staircase order is nonzero, so the identity would have to have coefficient $0$ on that monomial, and after renaming variables this contradicts monicity.\nFor part 1 there are several proofs: the original combinatorial one, a short proof by Rosset that works in the [[exterior algebra|0MDMZ965VE]], and proofs based on the [[Cayley–Hamilton theorem|0F765KFC2J]].\n\nExample: for $n = 1$, part 1 says $ab - ba = 0$ for scalars $a, b$, which is commutativity.\nFor $n = 2$, it says that the alternating sum of the $24$ products of four $2 \\times 2$ matrices in all orders is $0$.\n\nThe degree $2n$ is the best possible by part 2: $M_n(K)$ does not satisfy $s_{2n-1}$.\n\n## Explanations\n\n### L1 human\n\nMatrix multiplication is not commutative, but it is not completely free either.\nTake any $2n$ matrices of size $n \\times n$ and multiply them in all possible orders.\nAdd the products, each with the sign given by the parity of its order: the result is always the zero matrix.\nFor $1 \\times 1$ matrices this is just $ab = ba$.\n\n### L2 human\n\nThe theorem says that $M_n(K)$ is a [[polynomial identity ring|C0EDT81EGH]] with the standard identity $s_{2n}$, and that $2n$ is the minimal degree of an identity.\nThis fact is the base of the theory of PI rings, since by Kaplansky's and Posner's theorems many PI rings embed in matrix rings over commutative rings or fields.\nThe minimal degree $2n$ of an identity is then a measure of the size of the matrices needed.\n",
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