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  "body": "\n## Definition\n\nLet $k$ be a [[commutative ring|0310REJJY8]] with identity, and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $k$.\nLet $k\\langle x_1, \\ldots, x_n \\rangle$ be the [[free algebra|2P3Y2FRCSZ]] on $n$ variables.\nA polynomial $f \\in k\\langle x_1, \\ldots, x_n \\rangle$ is a *polynomial identity* of $A$ if $f(a_1, \\ldots, a_n) = 0$ for all $a_1, \\ldots, a_n \\in A$.\nThe polynomial $f$ is *monic* if some monomial of $f$ of highest degree has coefficient $1$.\nThe algebra $A$ is a *PI algebra* if it satisfies some monic polynomial identity.\nA [[ring|0B1261B5ED]] is a *polynomial identity ring*, or *PI ring*, if it is a PI algebra over $k = \\Z$.\n\n## Remarks\n\nOver a field $k$, every nonzero polynomial is a scalar multiple of a monic one, so a $k$-algebra is PI when it satisfies some nonzero polynomial identity.\nMonic polynomials are used for rings because, for example, every ring of characteristic $2$ satisfies the identity $2 x_1 = 0$, which says nothing about its multiplication.\n\nExample: a commutative ring satisfies the monic identity $x_1 x_2 - x_2 x_1$.\n\nExample: the ring $M_2(k)$ of $2 \\times 2$ matrices over a commutative ring $k$ satisfies the monic identity $[[x_1, x_2]^2, x_3]$, where $[a, b] = ab - ba$.\nIndeed, $c = [a_1, a_2]$ has trace $0$, so the Cayley–Hamilton theorem gives $c^2 = -\\det(c) \\, I$, a scalar matrix that commutes with every $a_3$.\n\nNon-example: let $k$ be a field and $A = k\\langle x, y \\rangle$.\nIf $f(x_1, \\ldots, x_n)$ is nonzero, then $f(x, xy, x y^2, \\ldots, x y^{n-1}) \\ne 0$, because distinct words in the $x y^i$ are distinct words in $x$ and $y$.\nSo the free algebra on two letters satisfies no nonzero identity and is not PI.\n\nEvery subring and every homomorphic image of a PI ring is a PI ring.\nThe [[exterior algebra|0MDMZ965VE]] of an infinite-dimensional vector space over a field of characteristic $0$ is PI, since it satisfies $[[x_1, x_2], x_3]$, but it satisfies no standard identity.\n",
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