Let R be a prime ring, Q the Utumi quotient ring of R and Ω the expansion closed set of products of automorphisms and skew derivations of Q. Let $\mathrm{\Lambda }\subseteq \mathrm{\Omega }\setminus {\mathrm{\Omega }}_m$, where $m\ge 0$, be such that for any $w\in \mathrm{\Lambda }(g,h)$, where $g,h$ are automorphisms of Q, $w(xy)-g(x)w(y)-w(x)h(y)$ is a sum of terms with $\delta (x)$ and ${\delta }^{\prime }(y)$, where $\delta \in {\mathrm{\Lambda }}_{|w|-1}$ or ${\delta }^{\prime }\in {\mathrm{\Lambda }}_{|w|-1}$ or $\delta ,{\delta }^{\prime }\in {\mathrm{\Omega }}_m$ (Definition 3). Suppose that no monic linear identities with words in $\mathrm{\Lambda }\cup {\mathrm{\Omega }}_m$ involve words in Λ. We prove the following:

(i) Λ extends to a basis of Ω (Theorem 1).

(ii) If a generalized polynomial φ with words in $\mathrm{\Lambda }\cup {\mathrm{\Omega }}_m$ is an identity, then so is the generalized polynomial obtained from φ by replacing every occurrence of $\nu (x)$ in φ by an arbitrary new variable $y_{\nu ,x}$ for $\nu \in \mathrm{\Lambda }$ and variable x (Theorem 2).

Dedekind's Lemma asserts that distinct automorphisms ${\sigma }_i$ of a commutative field are independent maps over the field and hence, via linearization, ${\sigma }_i(x_j)$ in distinct variables $x_j$, considered as distinct maps for distinct ordered pair $(i,j)$, are transcendental over the field in the case of characteristic 0. (ii) above generalizes this to maps defined by $w\in \mathrm{\Lambda }$ with dependency and transcendency over the prime ring R.

设R为素环，Q为R的 Utumi 商环，Ω 为Q的自同构和偏导导数的展开闭集积。让$\mathrm{\Lambda }\subseteq \mathrm{\Omega }\setminus {\mathrm{\Omega }}_{米}$， 在哪里 $米\ge 0$, 使得对于任何 $瓦\in \mathrm{\Lambda }(G,H)$， 在哪里 $G,H$是Q 的自同构，$瓦(X是)-G(X)瓦(是)-瓦(X)H(是)$ 是项的总和 $\delta (X)$ 和 ${\delta }^{\prime }(是)$， 在哪里 $\delta \in {\mathrm{\Lambda }}_{|瓦|-1}$ 要么 ${\delta }^{\prime }\in {\mathrm{\Lambda }}_{|瓦|-1}$ 要么 $\delta ,{\delta }^{\prime }\in {\mathrm{\Omega }}_{米}$（定义 3）。假设没有带有词的单调线性恒等式$\mathrm{\Lambda }\cup {\mathrm{\Omega }}_{米}$涉及Λ中的词。我们证明以下几点：

(i) Λ 扩展到Ω 的基（定理1）。

(ii)如果一个广义多项式φ包含词$\mathrm{\Lambda }\cup {\mathrm{\Omega }}_{米}$是一个恒等式，那么通过替换每个出现的φ从φ获得的广义多项式也是恒等式$\nu (X)$在φ由任意新变量${是}_{\nu ,X}$ 为了 $\nu \in \mathrm{\Lambda }$和变量x（定理 2）。

戴德金引理断言不同的自同构 ${\sigma }_{\mathrm{一世}}$ 交换域的 是域上的独立映射，因此，通过线性化， ${\sigma }_{\mathrm{一世}}(X_j)$ 在不同的变量中 $X_j$, 被视为不同有序对的不同映射 $(\mathrm{一世},j)$, 在特征 0 的情况下在场上是先验的。(ii)上面将其推广到由定义的地图$瓦\in \mathrm{\Lambda }$具有对素环R 的依赖性和超越性。