I scoured what I could in the literature but I have yet to find the information that should be out there. Consider the property

(P1) Every local subalgebra can be embedded in a local ideal subalgebra

for a commutative algebra A.

For lack of sufficient reference, let us say that a commutative algebra A is *thematic* iff A has property (P1) and every subalgebra of A has property (P1).

I feel that my research may require me to develop necessary and/or sufficient conditions for a finite commutative algebra A of prime characteristic p to be a thematic algebra. As usual, any insight or direction in the literature would be greatly appreciated. Thanks.