Zero-product balanced algebras

TL;DR

This paper introduces zero-product balanced algebras, a generalization of zero-product determined algebras, and explores their structural properties, including their characterization in commutative cases and implications for algebra maps.

Contribution

It defines zero-product balanced algebras, establishes their key properties, and characterizes their structure in commutative and unital cases, extending prior concepts.

Findings

Surjective zero-product preserving maps are weighted epimorphisms.

Commutative, zero-product balanced algebras are generated by idempotents.

Unital, zero-product balanced algebras either have a character or are generated by nilpotents.

Abstract

We say that an algebra is zero-product balanced if $ab\otimes c$ and $a\otimes bc$ agree modulo tensors of elements with zero-product. This is closely related to but more general than the notion of a zero-product determined algebra of Bre\v{s}ar, Gra\v{s}i\v{c} and Ortega. Every surjective, zero-product preserving map from a zero-product balanced algebra is automatically a weighted epimorphism, and this implies that zero-product balanced algebras are determined by their linear and zero-product structure. Further, the commutator subspace of a zero-product balanced algebra can be described in terms of square-zero elements. We show that a semiprime, commutative algebra is zero-product balanced if and only if it is generated by idempotents. It follows that every commutative, zero-product balanced algebra is spanned by nilpotent and idempotent elements. We deduce a dichotomy for unital, zero-product balanced algebras: They either admit a character or are generated by nilpotents.