Binomial Rings#

Abstract

Let K be a field. A split basic finite-dimensional K-algebra with left quiver Γ is monomial if it can be represented as the path algebra KΓ modulo an ideal generated by paths in Γ, and binomial if it can be represented as KΓ modulo relations of the form p = λq, where λ ∈ K and p and q are paths in Γ. Burgess et al. [Burgess, W. D., Fuller, K. R., Green, E. L., Zacharia, D. (1993). Left monomial rings–A generalization of monomial algebras. Osaka J. Math. 30(3):543–558] define monomial rings, generalizing monomial algebras; we here take this generalization one step further, introducing and studying the class of binomial rings, which includes the classes of binomial algebras, monomial rings, and square-free rings. We further show that every binomial ring has canonically associated with it a binomial algebra, with which the ring shares many properties.