Let ℳ be a structure of a signature Σ. For any ordered tuple $$ \overline{a}=\left({a}_1,\dots, {a}_{\mathrm{n}}\right) $$ of elements of ℳ, $$ {\mathrm{tp}}^{\mathcal{M}}\left(\overline{a}\right) $$ denotes the set of formulas θ(x1, …, xn) of a first-order language over Σ with free variables x1, . . . , xn such that $$ \mathcal{M}\left|=\theta \left({a}_1,\dots, {a}_n\right)\right. $$. A structure ℳ is said to be strongly ω-homogeneous if, for any finite ordered tuples $$ \overline{a} $$ and $$ \overline{b} $$ of elements of ℳ, the coincidence of $$ {\mathrm{tp}}^{\mathcal{M}}\left(\overline{a}\right) $$ and $$ {\mathrm{tp}}^{\mathrm{M}}\left(\overline{b}\right) $$ implies that these tuples are mapped into each other (componentwise) by some automorphism of the structure ℳ. A structure ℳ is said to be prime in its theory if it is elementarily embedded in every structure of the theory Th (ℳ). It is proved that the integral group rings of finitely generated relatively free orderable groups are prime in their theories, and that this property is shared by the following finitely generated countable structures: free nilpotent associative rings and algebras, free nilpotent rings and Lie algebras. It is also shown that finitely generated non-Abelian free nilpotent associative algebras and finitely generated non-Abelian free nilpotent Lie algebras over uncountable fields are strongly ω-homogeneous.

中文翻译：

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素数和齐次环和代数

令 ℳ 是签名 Σ 的结构。对于任何有序元组 $$ \overline{a}=\left({a}_1,\dots, {a}_{\mathrm{n}}\right) $$ 的元素 ℳ, $$ {\mathrm{ tp}}^{\mathcal{M}}\left(\overline{a}\right) $$ 表示具有自由变量 x1 的 Σ 上的一阶语言的一组公式 θ(x1, …, xn)， . . . , xn 使得 $$ \mathcal{M}\left|=\theta \left({a}_1,\dots, {a}_n\right)\right。$$。如果对于 ℳ 的元素的任何有限有序元组 $$ \overline{a} $$ 和 $$ \overline{b} $$，$$ {\mathrm {tp}}^{\mathcal{M}}\left(\overline{a}\right) $$ 和 $$ {\mathrm{tp}}^{\mathrm{M}}\left(\overline{b }\right) $$ 暗示这些元组通过结构ℳ的某种自同构相互映射（组件方式）。如果一个结构 ℳ 基本嵌入理论 Th (ℳ) 的每个结构中，则该结构 ℳ 在其理论中是素数。证明了有限生成的相对自由有序群的积分群环在它们的理论中是质数，并且这个性质为以下有限生成的可数结构所共有：自由幂零结合环和代数、自由幂零环和李代数。还表明有限生成的非阿贝尔自由幂零关联代数和有限生成的非阿贝尔自由幂零李代数在不可数域上是强ω-齐次的。并且该性质由以下有限生成的可数结构共享：自由幂零关联环和代数、自由幂零环和李代数。还表明有限生成的非阿贝尔自由幂零关联代数和有限生成的非阿贝尔自由幂零李代数在不可数域上是强ω-齐次的。并且该性质由以下有限生成的可数结构共享：自由幂零关联环和代数、自由幂零环和李代数。还表明有限生成的非阿贝尔自由幂零关联代数和有限生成的非阿贝尔自由幂零李代数在不可数域上是强ω-齐次的。