We give a simple characterization for a nonassociative algebra A, having characteristic 6= 2, to be commutative. Namely, A is commutative if and only if it is exible with a commuting set of generators. A counterexample shows that characteristic 6= 2 is necessary. Both the characterization and the counterexample were discovered using the computer algebra system in [2].

Let H be a Hopf algebra with a bijective antipode over a commutative ring k with unit. The Brauer group of H is defined as the Brauer group of Yetter–Drinfel’d H-module algebras, which generalizes the Brauer–Long group of a commutative and cocommutative Hopf algebra and those known Brauer groups of structured algebras.

An r-commutative algebra is an algebra A equipped with a Yang-Baxter operator R:A ⊗ A → A ⊗ A satisfying m = mR, where m:A ⊗ A → A is the multiplication map, together with the compatibility conditions R(a⊗ 1) = 1 ⊗ a, R(1 ⊗ a) = a ⊗ 1, R(id ⊗m) = (m ⊗ id)R2R1 and R(m ⊗ id) = (id ⊗ m)R1R2. The basic notions of differential geometry extend from commutative (or supercommutative) algebras to r-comm...

Abstract The purpose of this paper and its sequel is to develop the geometry built from commutative algebras that naturally appear as homology differential graded and, more generally, homotopy in spectra. question are those symmetric monoidal category abelian groups, being commutative, they form affine building blocks a geometry, rings algebraic geometry. We name Dirac because grading exhibits ...

Overview. My work is in the area of commutative algebra, and is motivated by the connections between algebra and geometry. Commutative algebra is the study of commutative rings (e.g. polynomial rings over fields and their quotients) and modules over these rings, while algebraic geometry is the study of finitely many polynomial equations in finitely many unknowns. The set of all solutions to suc...