Affine representability and decision procedures for commutativity theorems for rings and algebras

Representability of affine algebras over an arbitrary field

In a series of papers, we used full quivers as tools in describing PI-varieties of algebras and providing a complete proof of Belov's solution of Specht's problem for affine algebras over an arbitrary Noetherian ring. In this paper, utilizing ideas from that work, we give a full exposition of Belov's theorem that relatively free affine PI-algebras over an arbitrary field are representable. (Kemer proved the theorem over an infinite field.)

Representability and Specht problem for G-graded algebras

Let W be an associative PI algebra over a field F of characteristic zero, graded by a finite group G. Let id_{G}(W) denote the T-ideal of G-graded identities of W. We prove: 1. {[G-graded PI equivalence]} There exists a field extension K of F and a finite dimensional Z/2ZxG-graded algebra A over K such that id_{G}(W)=id_{G}(A^{*}) where A^{*} is the Grassmann envelope of A. 2. {[G-graded Specht problem]} The T-ideal id_{G}(W) is finitely generated as a T-ideal. 3. {[G-graded PI-equivalence for affine algebras]} Let W be a G-graded affine algebra over F. Then there exists a field extension K of…

Kemer's Theorem for Affine PI Algebras Over a Field of Characteristic Zero

We present a proof of Kemer's representability theorem for affine PI algebras over a field of characteristic zero.

诺特环上相对自由仿射代数的可表示性

Representability of relatively free affine algebras over a Noetherian ring

本文通过将提升方法从齐次多项式推广到非齐次多项式，给出了诺特交换环上仿射相对自由 PI 代数可表示性的新完整证明，并推广到对合、分次及非结合情形。

有限生成顶点代数的自同构群的可表示性

Representability of the automorphism group of finitely generated vertex algebras

研究自由代数的自同构群，证明有限生成顶点代数在其noetherian环上的自同构群为仿射群方案。

Hopfian and Bassian algebras

A ring $A$ is called Hopfian if $A$ cannot be isomorphic to a proper homomorphic image $A/J$. $A$ is called Bassian if there cannot be an injection of $A$ into a proper homomorphic image $A/J$. We consider classes of Hopfian and Bassian rings, and tie representability of algebras and chain conditions on ideals to these properties. In particular, any semiprime algebra satisfying the ACC on semiprime ideals is Hopfian, and any semiprime affine PI-algebra over a field is Bassian.