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        "quote": "并通过线性性延拓到整个 ${ \\mathcal { F } } ( V )$。分配律 (6.1) 自动得到满足，使得带有此乘积结构的 ${ \\mathcal { F } } ( V )$ 成为一个结合代数。代数 ${ \\mathcal { F } } ( V )$ 本质上没有对其施加任何“额外规则”，除了来自 $V$ 的元素的简单并置以及标量乘法。因此它被称为 $V$ 上的 <u class=\"concept-term\" data-concept-identity=\"urn:uuid:48420078-611a-40d6-b7c5-02935856b181\" data-concept-route=\"/concept-identities#concept-48420078-611a-40d6-b7c5-02935856b181\" id=\"first-67ed67400db27e00e05e-zh\">自由结合代数(free associative algebra)</u>。所有结合代数都可以构造为向量空间上自由结合代数的商代数。下面的例子说明了这一点。",
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        "quote": "and extend to all of ${ \\mathcal { F } } ( V )$ by linearity. The distributive law (6.1) is automatically satisfied, making ${ \\mathcal { F } } ( V )$ with this product structure into an associative algebra. The algebra ${ \\mathcal { F } } ( V )$ has in essence no ‘extra rules’ imposed on it other than simple juxtaposition of elements from $V$ and multiplication by scalars. It is therefore called the <u class=\"concept-term\" data-concept-identity=\"urn:uuid:48420078-611a-40d6-b7c5-02935856b181\" data-concept-route=\"/concept-identities#concept-48420078-611a-40d6-b7c5-02935856b181\" id=\"first-67ed67400db27e00e05e-en\">free associative algebra</u> over $V$. All associative algebras can be constructed as a factor algebra of the free associative algebra over a vector space. The following example illustrates this point.",
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