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 Ai−1. 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.
Bibliographical noteFunding Information:
Both authors are affiliated with the Institute of Mathematics, Hebrew University, Jerusalem, Israel. Sercombe was supported by a postdoctoral fellowship from ISF grant 686/17 of Shalev. Shalev was partially supported by ISF grant 686/17 and the Vinik Chair of Mathematics, which he holds. MSC2020: primary 16P70; secondary 16P10. Keywords: associative algebras, length, depth, chain conditions.
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- associative algebras
- chain conditions