Let $\Gamma$ be the $T$-ideal of identities of an affine PI-algebra over an algebraically closed field $F$ of characteristic zero. Consider the family ${\mathcal{ℳ}}_{\Gamma }$ of finite dimensional algebras $\mathrm{\Sigma }$ with $\mathrm{Id}<!--FUNCTION\; APPLICATION-->(\mathrm{\Sigma })=\Gamma$. By Kemer’s theory ${\mathcal{ℳ}}_{\Gamma }$ is not empty. We show there exists $A\in {\mathcal{ℳ}}_{\Gamma }$ with Wedderburn–Malcev decomposition $A\cong <!--FUNCTION\; APPLICATION-->A_{ss}\oplus J_A$, where $J_A$ is the Jacobson’s radical and $A_{ss}$ is a semisimple supplement with the property that if $B\cong <!--FUNCTION\; APPLICATION-->B_{ss}\oplus J_B\in {\mathcal{ℳ}}_{\Gamma }$ then $A_{ss}$ is a direct summand of $B_{ss}$. In particular $A_{ss}$ is unique minimal, thus an invariant of $\Gamma$. More generally, let $\Gamma$ be the $T$-ideal of identities of a PI algebra and let ${\mathcal{ℳ}}_{{\mathbb{Z}}_2,\Gamma }$ be the family of finite dimensional superalgebras $\mathrm{\Sigma }$ with $\mathrm{Id}<!--FUNCTION\; APPLICATION-->(E(\mathrm{\Sigma }))=\Gamma$. Here $E$ is the unital infinite dimensional Grassmann algebra and $E(\mathrm{\Sigma })$ is the Grassmann envelope of $\mathrm{\Sigma }$. Again, by Kemer’s theory ${\mathcal{ℳ}}_{{\mathbb{Z}}_2,\Gamma }$ is not empty. We prove there exists a superalgebra $A\cong <!--FUNCTION\; APPLICATION-->A_{ss}\oplus J_A\in {\mathcal{ℳ}}_{{\mathbb{Z}}_2,\Gamma }$ such that if $B\in {\mathcal{ℳ}}_{{\mathbb{Z}}_2,\Gamma }$, then $A_{ss}$ is a direct summand of $B_{ss}$ as superalgebras. Finally, we fully extend these results to the $G$-graded setting where $G$ is a finite group. In particular we show that if $A$ and $B$ are finite dimensional $G_2:={\mathbb{Z}}_2\times G$-graded simple algebras then they are $G_2$-graded isomorphic if and only if $E(A)$ and $E(B)$ are $G$-graded PI-equivalent.

让$\gamma$成为$\mathrm{时间}$- 代数闭域上仿射 PI 代数的理想恒等式$F$的特征为零。考虑家庭${\mathcal{ℳ}}_{\gamma }$有限维代数$\mathrm{\Sigma }$和$\mathrm{ID}<!--FUNCTION\; APPLICATION-->（\mathrm{\Sigma }）=\gamma$。根据凯梅尔的理论${\mathcal{ℳ}}_{\gamma }$不为空。我们证明存在$A\epsilon {\mathcal{ℳ}}_{\gamma }$韦德伯恩-马尔切夫分解$A\cong <!--FUNCTION\; APPLICATION-->A_{ss}\oplus J_A$， 在哪里$J_A$是 Jacobson 的激进式，$A_{ss}$是一个半简单的补充，其性质是如果$乙\cong <!--FUNCTION\; APPLICATION-->{乙}_{ss}\oplus J_{乙}\epsilon {\mathcal{ℳ}}_{\gamma }$然后$A_{ss}$是一个直接求和${乙}_{ss}$。尤其$A_{ss}$是唯一最小的，因此是一个不变量$\gamma$。更一般地，让$\gamma$成为$\mathrm{时间}$- PI 代数的理想恒等式，令${\mathcal{ℳ}}_{{\mathbb{Z}}_2,\gamma }$是有限维超代数族$\mathrm{\Sigma }$和$\mathrm{ID}<!--FUNCTION\; APPLICATION-->（乙（\mathrm{\Sigma }））=\gamma$。这里$乙$是单位无限维格拉斯曼代数，$乙（\mathrm{\Sigma }）$是格拉斯曼包络$\mathrm{\Sigma }$。再次，根据凯梅尔的理论${\mathcal{ℳ}}_{{\mathbb{Z}}_2,\gamma }$不为空。我们证明存在一个超代数$A\cong <!--FUNCTION\; APPLICATION-->A_{ss}\oplus J_A\epsilon {\mathcal{ℳ}}_{{\mathbb{Z}}_2,\gamma }$这样如果$乙\epsilon {\mathcal{ℳ}}_{{\mathbb{Z}}_2,\gamma }$， 然后$A_{ss}$是一个直接求和${乙}_{ss}$作为超代数。最后，我们将这些结果完全推广到$G$-分级设置，其中$G$是一个有限群。我们特别证明如果$A$和$乙$是有限维的$G_2：={\mathbb{Z}}_2\times G$-分级简单代数然后它们是$G_2$- 分级同构当且仅当$乙（A）$和$乙（乙）$是$G$- 分级 PI 等效值。