# Smallest generating set for a commutative algebra

Consider a commutative algebra of $$w \times w$$ matrices over a field $$\mathbb{F}$$, $$A \subset M_{w}(\mathbb{F})$$. Let $$S = \{M_1,\ldots,M_t\}$$ be a minimal generating set for $$A$$. That is (a) the smallest ring that contains $$S$$ is exactly $$A$$, and (b) no subset of $$S$$ generates all of $$A$$.

I want to view $$A = \mathbb{F}[M_1,\ldots,M_t]$$ as a quotient ring $$R = \mathbb{F}[x_1,\ldots,x_t]/I$$, by associating $$M_i$$ with $$x_i$$ and taking $$I$$ as the ideal of dependencies between $$M_1,\ldots,M_t$$.

What can be said about the "number of variables" in the quotient ring ($$t=|S|$$)? In particular, I want to know the following.

1. Is there a tighter (than $$t \leq w$$) upper bound for $$t$$ that possibly depends on more properties of $$A$$? Could there be another $$R' = \mathbb{F}[y_1,\ldots,y_{t'}]/J$$ analogous to $$A$$?

• For example, the rings $$R_1 = \mathbb{F}[x_1,x_2]/\langle x_1^2 - x_2, x_2^2 \rangle$$ and $$R_2 = \mathbb{F}[y]/\langle y^4 \rangle$$ essentially yield the same matrix algebra. Is there a way to look at a ring and determine that it is "sub-optimal" like the above $$R_1$$?
2. When a set $$T$$ generates $$A$$, but is not minimal, what property does it violate? For example, a spanning set for a vector space is minimal (a basis) if and only if it is a linearly independent set.

• I think I am looking for some sort of "algebraic dependence modulo the ideal I". Sorry, I wish I could be more precise here.

For both parts, if the answer depends on the field $$\mathbb{F}$$, then in what ways does it do so?

P.S.: The edits have been made after looking into a reference suggested by @Wuestenfux in a comment on an older post.

• I'd look into Groebner bases for quotient rings. You will find this in the book ''Using Algebraic Geometry'' by Cox et al. Sep 20, 2019 at 12:17
• @Wuestenfux: Thanks a lot for the reference. I have solved the question. Although, I don't know if I should remove the question or post an answer here. Sep 23, 2019 at 5:27