The length and depth of associative algebras

Recently there has been considerable interest in studying the length and depth of finite groups, algebraic groups Lie groups. In this paper we introduce study similar notions for algebras. Let $k$ be a field let $A$ an associative, not necessarily unital, algebra over $k$. An unrefinable chain is subalgebras $A=A_0>A_1>...>A_t=0$ some integer $t$ where each $A_i$ maximal subalgebra $A_{i-1}$. The (respectively, minimal) such called depth) $A$. It turns out that length, dimension are equivalent properties For dimensional, give formula $A$, bound when equals its respectively. Finally, investigate under what circumstances bounded above by function or depth, minus depth.