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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity.\nFor a subset $X \\subseteq R$, the *right annihilator* of $X$ is $r(X) = \\set{a \\in R : x a = 0 \\text{ for all } x \\in X}$, a right ideal of $R$.\nThe ring $R$ is *right Goldie* if:\n\n1. $R$ has finite right uniform dimension: there is no infinite family of nonzero right ideals $I_1, I_2, \\ldots$ of $R$ whose sum is direct;\n2. $R$ satisfies the [[ascending chain condition|0GZ6Y1H4RN]] on right annihilators: every increasing chain $r(X_1) \\subseteq r(X_2) \\subseteq \\cdots$ is eventually constant.\n\nLeft Goldie rings are defined in the same way with left ideals and left [[annihilators|HB4XDKJTW8]].\nA ring is *Goldie* if it is both right and left Goldie.\n\n## Remarks\n\nExample: a right [[Noetherian ring|0670T59HGE]] $R$ is right Goldie.\nAn infinite direct sum $I_1 \\oplus I_2 \\oplus \\cdots$ of nonzero right ideals would give a strictly increasing chain $I_1 \\subset I_1 \\oplus I_2 \\subset \\cdots$, which is impossible.\nRight annihilators are right ideals, so they satisfy the ascending chain condition.\n\nExample: every commutative integral domain is Goldie, since two nonzero ideals $I, J$ always meet in the nonzero ideal $IJ$, and the only annihilators are $0$ and $R$.\n\nNon-example: let $k$ be a field and $R = \\prod_{n \\ge 1} k$.\nThe ideals $I_n = k e_n$, where $e_n$ has $1$ in coordinate $n$ and $0$ elsewhere, are nonzero and their sum is direct, so clause 1 fails.\n\nClause 1 can be stated with [[uniform modules|YXEVE39Q31]]: $R_R$ has finite uniform dimension exactly when it contains an essential submodule that is a finite direct sum of uniform right ideals.\n",
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