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  "body": "\n## Definition\n\nLet $A$ be a [[commutative ring|0310REJJY8]].\nA [[topology|0CQHFPGJ45]] on $A$ is *linear* if it satisfies both conditions:\n\n1. With this topology, $A$ is a topological ring.\n2. The open [[ideals|096Z9MP2ME]] of $A$ form a fundamental system of neighbourhoods of $0$: every neighbourhood of $0$ contains an open ideal.\n\nMore generally, let $M$ be an [[$A$-module|0NZ8JA25WK]] and give $A$ a linear topology.\nA topology on $M$ is *linear* if $M$ is a topological group under addition and the scalar multiplication $A \\times M \\to M$ is continuous.\nIn addition, the open [[submodules|05WDSN597C]] of $M$ must form a fundamental system of neighbourhoods of $0$.\n\n## Remarks\n\nEquivalently, a linear topology on $A$ is given by a family $\\mathcal{F}$ of ideals such that each intersection $I \\cap J$ of two members contains a member.\nA set $U$ is then open when for each $x \\in U$ there is $I \\in \\mathcal{F}$ with $x + I \\subseteq U$.\n\nExample: for an ideal $I$ of $A$, the [[I-adic topology|2VF2TSJ3PY]] has the powers $I^n$ as a fundamental system of neighbourhoods of $0$.\nEach $I^n$ is an open ideal, so clause 2 holds; addition and multiplication are continuous because $(x + I^n) + (y + I^n) = x + y + I^n$ and $(x + I^n)(y + I^n) \\subseteq xy + I^n$.\nThe discrete topology is linear, with the zero ideal open.\n\nNon-example: the usual topology on $\\R$ makes $\\R$ a topological ring, so clause 1 holds.\nBut the only ideals of the field $\\R$ are $0$ and $\\R$.\nThe set $\\set{0}$ is not open, and the interval $(-1, 1)$ does not contain $\\R$; so clause 2 fails.\n",
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