The lengths of matrix incidence algebras are studied when their radicals have square zero. All realizable values of the length function are provided for such algebras. In order to obtain this result, a discrete optimization problem is posed and solved. Also, the exact formula of the length is deduced under the additional assumption that the algebra is maximal by inclusion. Moreover, the solution to the length realizability problem is established for matrix incidence algebras with arbitrary radicals under a restriction on the cardinality of the ground field.

中文翻译：

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关于根为零的矩阵关联代数的长度

研究了当其根式为零时矩阵关联代数的长度。为此类代数提供了长度函数的所有可实现值。为了获得这个结果，提出并解决了一个离散优化问题。此外，长度的精确公式是在代数因包含而最大的附加假设下推导出来的。此外，在基场基数的限制下，建立了任意根矩阵关联代数的长度可实现性问题的解。