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  "body": "\n## Definition\n\nA *difference ring* is a pair $(R, \\sigma)$, where $R$ is a commutative [[ring|0B1261B5ED]] with identity and $\\sigma \\colon R \\to R$ is a [[ring homomorphism|0VTH8KPCG2]], that is:\n\n1. $\\sigma(a + b) = \\sigma(a) + \\sigma(b)$ for all $a, b \\in R$;\n2. $\\sigma(a b) = \\sigma(a) \\sigma(b)$ for all $a, b \\in R$;\n3. $\\sigma(1) = 1$.\n\nA *difference field* is a difference ring $(R, \\sigma)$ in which $R$ is a [[field|04C5ZBG1WT]].\nA difference ring is *inversive* if $\\sigma$ is an automorphism.\n*Difference algebra* is the study of difference rings, difference fields, and the difference equations and ideals over them.\n\n## Remarks\n\nSome authors require $\\sigma$ to be an automorphism in the definition of a difference ring or field.\nA *difference polynomial* over $(R, \\sigma)$ is a polynomial in the variables $y, \\sigma y, \\sigma^2 y, \\dots$, and a solution is an element $a$ of a difference ring extension at which it vanishes.\n\nExample: $(\\C(x), \\sigma)$ with $\\sigma(f)(x) = f(x + 1)$ is an inversive difference field.\nThe shift is additive and multiplicative, it fixes $1$, and its inverse is $f(x) \\mapsto f(x - 1)$.\n\nExample: a field $K$ of characteristic $p \u003e 0$ with the [[Frobenius map|0T3WVTD675]] $\\sigma(a) = a^p$ is a difference field.\n\nNon-example: $(\\Z, \\sigma)$ with $\\sigma(n) = 2n$ is not a difference ring.\nIt satisfies clause 1, but $\\sigma(1) = 2 \\ne 1$, so clause 3 fails, and $\\sigma(1 \\cdot 1) = 2 \\ne 4 = \\sigma(1)\\sigma(1)$, so clause 2 fails too.\n",
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