Let $R$ be a commutative ring with identity, $X$ be an indeterminate and $I_c(R)$ be the set of elements $a$ of $R$ such that there exists an $R$-homomorphism of rings $\sigma :R[[X]]\to R$ with $\sigma (X)=a$. O’Malley called $R$ to be power invariant (respectively, strongly power invariant) if whenever $S$ is a ring such that $R[[X]]$ is isomorphic to $S[[X]]$ (respectively, whenever $S$ is a ring and $\phi$ is an isomorphism of $R[[X]]$ onto $S[[X]]$), then $R$ and $S$ are isomorphic (respectively, then there exists an $S$-automorphism $\psi$ of $S[[X]]$ such that $\psi (X)=\phi (X)$) [M. O’Malley, Isomorphic power series rings, Pacific J. Math. 41(2) (1972) 503–512]. We prove that a ring $R$ is power invariant in each of the following case: $(1)$$R$ is a domain in which $I_c(R)$ is comparable to each radical ideal of $R$ (for instance a domain with Krull dimension one), $(2)$$R$ is a domain in which Jac$(R)$ (i.e. the Jacobson radical of $R$) is comparable to each radical ideal of $R$ and $(3)$$R$ is a Prüfer domain. Also in each of the aforementioned case, we prove that either $R$ is strongly power invariant or $R$ is isomorphic to a quasi-local power series ring. Let $M$ be a unital module over $R$. We show that if $R$ is reduced and strongly power invariant, then Nagata’s idealization ring $R(+)M$ is strongly power invariant (but the converse is false). Ishibashi called a ring $R$ to be strongly$n$-power invariant if whenever $S$ is a ring and $\phi$ is an isomorphism of $R[[X_1,\dots ,X_n]]$ onto $S[[X_1,\dots ,X_n]]$, then there exists an $S$-automorphism $\psi$ of $S[[X_1,\dots ,X_n]]$ such that $\psi (X_i)=\phi (X_i)$ for each $i$. We prove that if $R$ is a ring in which $I_c(R)$ is nil, then $R$ is strongly$n$-power invariant for all positive integer $n$. We deduce that every polynomial ring in finitely many indeterminates is strongly$n$-power invariant for all positive integer $n$.

让$R$是一个具有恒等式的交换环，$X$是一个不确定的和${我}_C(R)$是元素的集合$\mathrm{一个}$的$R$使得存在一个$R$- 环的同态$\sigma ：R[[X]]\to R$和$\sigma (X)=\mathrm{一个}$. 奥马利打电话给$R$是幂不变的（分别是强幂不变的），如果$\mathrm{小号}$是一个环，使得$R[[X]]$同构于$\mathrm{小号}[[X]]$（分别地，每当$\mathrm{小号}$是一个环并且$\phi$是一个同构$R[[X]]$到$\mathrm{小号}[[X]]$）， 然后$R$和$\mathrm{小号}$是同构的（分别存在$\mathrm{小号}$-自同构$\psi$的$\mathrm{小号}[[X]]$这样$\psi (X)=\phi (X)$) [M. O'Malley，同构幂级数环，Pacific J. Math。 41 (2) (1972) 503–512]。我们证明一个环$R$在以下每种情况下都是幂不变的：$(1)$$R$是一个域，其中${我}_C(R)$可与每一个激进的理想相媲美$R$（例如具有 Krull 维一的域），$(2)$$R$是一个域，其中 Jac$(R)$（即雅各布森根$R$) 可与$R$和$(3)$$R$是一个 Prüfer 域。同样在上述每种情况下，我们证明$R$是强幂不变的或$R$同构于准局部幂级数环。让$米$是一个单元模块$R$. 我们证明如果$R$是约简和强幂不变性，则 Nagata 的理想化环$R(+)米$是强幂不变的（但反过来是错误的）。石桥叫响$R$要坚强$n$- 幂不变，如果任何时候$\mathrm{小号}$是一个环并且$\phi$是一个同构$R[[X_1,\dots ,X_n]]$到$\mathrm{小号}[[X_1,\dots ,X_n]]$, 那么存在一个$\mathrm{小号}$-自同构$\psi$的$\mathrm{小号}[[X_1,\dots ,X_n]]$这样$\psi (X_{\mathrm{一世}})=\phi (X_{\mathrm{一世}})$对于每个$\mathrm{一世}$. 我们证明如果$R$是一个环，其中${我}_C(R)$为零，则$R$是强烈的$n$-所有正整数的幂不变$n$. 我们推导出有限多不定项中的每个多项式环是强$n$-所有正整数的幂不变$n$.