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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nA left ideal $I$ of $R$ is principal if there is an element $a \\in R$ such that $I = (a)$, the [[left ideal generated by|030TZ0DQWT]] $\\{a\\}$.\n\n## Remarks\n\nThe left ideal generated by $a$ is $R a = \\{r a : r \\in R\\}$.\nIf $R$ is commutative, the principal ideal $(a)$ is the set of the multiples of $a$.\nEvery ideal of $\\mathbb{Z}$ is principal: each one is $n\\mathbb{Z}$ for some integer $n \\ge 0$.\nIn $\\mathbb{Z}[x]$, the ideal $(2, x)$ is not principal.\n\n## Explanations\n\n### L1 human\n\nA principal ideal is the set of all the multiples of one element.\nFor example, the multiples of $3$ in the integers form the principal ideal $(3)$.\n",
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