Алгебра с единицей

Алгебра с единицей (также унитальная алгебра, калька с англ. unital algebra) — алгебра над кольцом, в которой существует нейтральный элемент по отношению к умножению — единица, то есть такой элемент $1$ , что для всех элементов $x$ алгебры выполняются равенства:

1\cdot x=x\cdot 1=x

Это определение эквивалентно тому, что данная алгебра является моноидом по отношению к умножению. Как и в случае любого моноида, нейтральный элемент является единственным.

Многие ассоциативные алгебры, включая алгебры групп, многочленов и матриц обладают единицей, если нейтральным элементом обладают соответствующие кольца. Большинство функциональных алгебр, рассматриваемых в анализе, напротив, единицей не обладают. К таковым относится, например, алгебра функций с интегрируемым квадратом и неограниченной областью определения, а также алгебра функций, являющихся бесконечно малыми на бесконечности (особенно функций с компактным носителем в некоторых некомпактных пространствах).

Литература

Kevin McCrimmon. A taste of Jordan algebras. — Springer, 2004. — ISBN 978-0-387-95447-9.
Скорняков Л. А., Шестаков И. П. . Глава III. Кольца и модули // Общая алгебра / Под общ. ред. Л. А. Скорнякова. — М.: Наука, 1990. — Т. 1. — С. 291—572. — 592 с. — (Справочная математическая библиотека). — 30 000 экз. — ISBN 5-02-014426-6.

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