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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] and let $\\mathcal{T}$ be a [[topology|0CNQXVBHCX]] on the set $R$.\nGive $R \\times R$ the [[product topology|0Z57EC6YX3]].\nThe pair $(R, \\mathcal{T})$ is a *topological ring* if the following maps are [[continuous|0SY9ECQ19Q]]:\n\n1. addition $R \\times R \\to R$, $(a, b) \\mapsto a + b$;\n2. negation $R \\to R$, $a \\mapsto -a$;\n3. multiplication $R \\times R \\to R$, $(a, b) \\mapsto ab$.\n\n## Remarks\n\nClauses 1 and 2 say that $(R, +)$ is a topological group.\nExample: $\\R$ with the usual topology; continuity of $(a, b) \\mapsto ab$ follows from $ab - a'b' = a(b - b') + (a - a')b'$.\nExample: any ring with the discrete topology, since every map from a discrete space is continuous.\nExample: for an ideal $I$ of a commutative ring $R$, the $I$-adic topology, with the ideals $I^n$ as a base of neighbourhoods of $0$.\nNon-example: $\\R$ with the lower limit topology, whose base is the intervals $[a, b)$.\nNegation is not continuous: the preimage of the open set $[0, 1)$ is $(-1, 0]$, which is not open there.\n",
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