sdg: quotients and fermat-hadamard rings

[stollluk]

Some material on quotients by congruence relations over endomorphism theories and the relationship with ring-theoeretic ideals.

[stollluk]

Given an extension T of the theory of rings, every T-congruence relation on a T-algebra A induces an ideal of the
underlying ring of A. It is well-known that the converse holds if T is a Fermat theory. Maybe one could extend this to a
characterization of Fermat theories? At least the existence of Hadamard quotients (as opposed to their uniqueness)
follows from the assumption that every ideal on the underlying ring of a T-algebra induces a T-congruence.

[felixwellen]

The statement alluded to in the comment above is prop. 1.9 in the sdg notes (state of this PR/commit).

[markrd-williams]

The property you suggested, regularity of generators of free algebras seems reasonable to me.

The property of $X-Y$ being regular in $F(n+2)$ reduces to $X$ being regular in $F(n+2)$, by an automorphism swapping $X$ and $X-Y$. Further it is enough to ask for $X$ being regular in $F(1)$ to get this property.

This reads as some sort of continuity condition to me, if the base ring is a field, then $X$ regular says: for all $f : R \to R$ s.t $f(r) = 0$ for all $r \neq 0$ then $f = 0$.