Class Of Commutative Algebras Research Articles

Isomorphism classes of commutative algebras generated by idempotents

We study commutative algebras generated by idempotents with particular emphasis on the number of primitive idempotents. Let [Formula: see text] be an integral domain with the field of fractions [Formula: see text] and let [Formula: see text] be an [Formula: see text]-algebra which is torsion-free as an [Formula: see text]-module. We show that if [Formula: see text] satisfies the three conditions: [Formula: see text] is generated by idempotents over [Formula: see text]; [Formula: see text] is countably infinite dimensional over [Formula: see text]; [Formula: see text] has [Formula: see text] primitive idempotents for a nonnegative integer [Formula: see text], then [Formula: see text] is uniquely determined up to [Formula: see text]-algebra isomorphism. We also consider the case where [Formula: see text] has countably many primitive idempotents.

On context-equivalence of algebras

The notion of context-equivalent algebras introduced by Müller generalizes Morita-equivalence. It is a coarser equivalence in the class of algebras but it still preserves many ring-theoretic properties. It also gives a new equivalence in the class of commutative algebras. We present a version of this equivalence which includes nonunital algebras. In particular, it allows one to relate properties of an algebra and its ideals. We give a criterion for context-equivalent algebras in terms of finite-rank and adjointable operators. Further, we find a better criterion for equivalence for commutative algebras in terms of ideals.

On Trace Forms on a Class of Commutative Algebras Satisfying an Identity of Degree Four

In this paper, we deal with commutative algebras A satisfying the identity 2β{(xy)2 - x2y2} + γ{((xy)x)y + ((xy)y)x - (y2x)x - (x2y)y} = 0, where β, γ are scalars. These algebras appeared as one of the four families of degree four identities in Carini, Hentzel and Piacentini-Cattaneo [2]. We prove that if the algebra A admits an identity element, then A is associative. We also prove that there exist trace forms on A. Finally, we prove that if A has a non-degenerate trace form, then A satisfies the identity ((yx)x)x = y((xx)x), a generalization of right alternativity. Our results require characteristic ≠ 2, 3.

Cluster algebras I: Foundations

In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.