  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.
