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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.

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This is actually a lot simpler than I thought. Firstly, for finite commutative algebras the (P1) property and the more general thematic property are equivalent. Moreover, using the fact that any such algebra can be uniquely decomposed as a direct sum of local algebras, it becomes clear that a finite commutative algebra A has property (P1) iff A admits at most one local subalgebra with non-trivial nilradical.

If we take into consideration possibly infinite commutative algebras, what might this necessary and sufficient condition become ?

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That is, the simplification given above is true for finite commutative algebras of prime characteristic. – Oliver Kayende Jan 26 '13 at 5:52

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