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
be a
field and let
be an associative, not necessarily unital, algebra over
. An unrefinable
chain of
is a chain
of subalgebras
for
some integer
, where
each
is a maximal
subalgebra of
. The
maximal (respectively, minimal) length of such an unrefinable chain is called the length (respectively,
depth) of
.
It turns out that finite length, finite depth and finite dimension are equivalent properties
for
.
For
finite dimensional, we give a formula for the length of
, we bound the depth
of
, and we study
when the length of
equals its dimension and its depth respectively. Finally, we investigate under what circumstances
the dimension of
is bounded above by a function of its length, or its depth, or its length minus its
depth.
