Prime and Homogeneous Rings and Algebras

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, …, x_n) of a first-order language over Σ with free variables x₁, . . . , x_n 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.