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Let $V$ be some finite-dimensional vector space (over some field $\mathbb{K}$), then a (possibly non-associative) algebra $A$ on $V$ corresponds to a bilinear map $V \times V \to V$. I prefer answers that can address this level of generality.

Given some (finite) subset $\{v_1, \dots, v_D\}$ of $A$, let $\mathscr{A}(v_1, \dots, v_D)$ denote the sub-algebra of $A$ generated by $\{v_1, \dots, v_D\}$, i.e. by applying the bilinear algebra product $V \times V \to V$ and linear combinations. ("Non-commutative and non-associative polynomials".)

Question: Is there a notion of "algebraic dimension" for such algebras which reflects the minimal number $D$ of elements of $A$ needed to generate $A$, i.e. the smallest possible $D$ such that there exists a $\{v_1, \dots, v_D\} \subset A$ with $\mathscr{A}(v_1, \dots, v_D) = A$?

What is the name of this notion of "algebraic dimension"? Any recommendations for references to read more about it?

Comments:

  • A necessary, but obviously in general not sufficient, condition for such a generating set is that it be linearly independent. Therefore, if $N$ is the vector space dimension of $A$ (i.e. of $V$), then clearly in general $D \le N$, where $D$ denotes this "algebraic dimension".

  • Also I think such a minimal $D$ has to exist e.g. by applying Zorn's lemma? But if such a concept doesn't exist in the literature because in general it doesn't exist or isn't well-defined, it would be helpful to explain why and/or provide a reference or counterexample.

  • If this is well-defined, I still wouldn't expect such minimal "algebraically spanning" sets to necessarily satisfy the matroid axioms, i.e. be as well behaved as the concept of linear independence. But you know that's a whole other can of worms.

What I already tried (i.e. similar concepts that aren't what I'm looking for):

  • In cases when the algebra is sufficiently "free", then the Krull dimension appears to coincide with this notion (cf. e.g. this question), or be off only by one, e.g. the polynomial algebra generated by two elements $\mathbb{K}[x,y]$ is $2$ (but maybe in my definition the value should be $3$ to include constant terms but anyway).

    Anyway the problem with Krull dimension for finite-dimensional algebras is that they all satisfy the descending chain condition, so they must have deviation zero, so they must be (left+right) Artinian rings, so necessarily have Krull dimension zero. So basically Krull dimension penalizes "non-freeness" too much to be useful for studying finite-dimensional algebras in this way.

  • "Algebras of finite type" and/or "algebraic independence" (cf. e.g. [1][2][3][4]). Both of these concepts necessarily assume commutative and associative algebras (basically so we can reduce things to quotients of polynomial algebras?). Also there doesn't appear to be a term for the minimal number of elements in an algebraically independent set. But anyway what I want I guess could be considered the generalization of this to the non-commutative, non-associative setting, so I guess anything that can be written as a quotient of a free non-associative algebra.

  • Dilworth number (cf. this very closely related question). The finite-dimensionality presumably makes all finite-dimensional algebras not only (left+right) Artinian rings (satisfying DCC) but also (left+right) Noetherian rings. (Apparently Artinian implies Noetherian even in the non-commutative case, so maybe this observation is redundant.) Again, this is ridiculously close to what I want, but again it assumes that things are commutative and associative and thus quotients of polynomial rings, which again is not an assumption that I want to make. Also the question is technically asking about ideals and not subrings (i.e. sub-algebras) - that may not be relevant here because $A$ is both a (non-proper) ideal and subring, but for what it's worth.

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