In this paper, we develop a small cancellation theory for associative algebras with a basis of invertible elements. Namely, we study quotients of a group algebra of a free group and introduce three axioms for the corresponding defining relations. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a linear space and establish the corresponding structure theorems. We also provide a revision of a concept of Gröbner basis for our rings and establish a greedy algorithm for the Ideal Membership Problem.

中文翻译：

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环的类群小抵消理论

在本文中，我们开发了一种基于可逆元素的关联代数的小型抵消理论。也就是说，我们研究自由群的群代数的商，并为相应的定义关系引入三个公理。我们证明所获得的环是不平凡的。此外，我们证明了该环具有符合关系的全局过滤，找到了该环作为线性空间的基础并建立了相应的结构定理。我们还对环的 Gröbner 基概念进行了修订，并为理想隶属问题建立了贪婪算法。