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arXiv · 2010.02836

Group-like small cancellation theory for rings

A. Atkarskaya·A. Kanel-Belov·E. Plotkin·E. Rips
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Abstract

In the present 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 vector space and establish the corresponding structure theorems. We also provide a revision of a concept of Gr\"{o}bner basis for our rings and establish a greedy algorithm for the Ideal Membership Problem.

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A. Atkarskaya, A. Kanel-Belov, E. Plotkin, E. Rips. 2020-10-06. Group-like small cancellation theory for rings. https://doi.org/10.1142/s0218196723500522

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