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The Lengths of Matrix Incidence Algebras Over Small Finite Fieldsстатья
Статья опубликована в журнале из списка Web of Science и/или Scopus
- Журнал: Journal of Mathematical Sciences
- Том: 262
- Номер: 1
- Год издания: 2022
- Издательство: Plenum Publishers
- Местоположение издательства: United States
- Первая страница: 62
- Последняя страница: 83
- DOI: 10.1007/s10958-022-05800-4
- Аннотация: The paper considers the problem of computing the lengths of matrix incidence algebras over a field whose cardinality is strictly less than the matrix order n. For n = 3, 4, the lengths of all such algebras over the field of two elements are determined. In the case where the ground field and the number n are arbitrary but the Jacobson radical of the algebra has nilpotency index 2, an upper bound for the length is provided. In addition, the incidence algebras isomorphic to a direct sum of triangular matrix algebras of order 2 and an algebra of diagonal matrices are considered. It is shown that the lengths of these algebras over the field of two elements can take only two distinct values, which can be determined exactly. Moreover, the diagonal number of a matrix incidence algebra is introduced and bounded above.