Zero-product balanced algebras

Autor: Eusebio Gardella, Hannes Thiel
Rok vydání: 2023
Předmět:
Zdroj: Linear Algebra and its Applications. 670:121-153
ISSN: 0024-3795
DOI: 10.1016/j.laa.2023.04.015
Popis: 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.
Comment: 25 pages. This is the published version
Databáze: OpenAIRE
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