Vol. 311, No. 1, 2021

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The length and depth of associative algebras

Damian Sercombe and Aner Shalev

Vol. 311 (2021), No. 1, 197–220
DOI: 10.2140/pjm.2021.311.197

Recently there has been considerable interest in studying the length and the depth of finite groups, algebraic groups and Lie groups. We introduce and study similar notions for algebras. Let k be a field and let A be an associative, not necessarily unital, algebra over k. An unrefinable chain of A is a chain of subalgebras A = A0 > A1 > > At = 0 for some integer t, where each Ai is a maximal subalgebra of Ai1. The maximal (respectively, minimal) length of such an unrefinable chain is called the length (respectively, depth) of A. It turns out that finite length, finite depth and finite dimension are equivalent properties for A. For A finite dimensional, we give a formula for the length of A, we bound the depth of A, and we study when the length of A equals its dimension and its depth respectively. Finally, we investigate under what circumstances the dimension of A is bounded above by a function of its length, or its depth, or its length minus its depth.

associative algebras, length, depth, chain conditions
Mathematical Subject Classification
Primary: 16P70
Secondary: 16P10
Received: 20 April 2020
Accepted: 19 January 2021
Published: 17 March 2021
Damian Sercombe
Department of Mathematics
Ruhr-Universität Bochum
Aner Shalev
Einstein Institute of Mathematics
The Hebrew University of Jerusalem