(I) Suppose A is a finite commutative local algebra. Must every lattice of local subalgebras of A be a distributive lattice ?
By a subalgebra of A we mean an algebra contained in A that shares the same unity element. By a lattice of subalgebras of A we mean, as usual, a family of subalgebras of A that is partially ordered with respect to set inclusion, and each of whose non-empty finite subsets admits an infimum and supremum. As usual, by a distributive lattice of subalgebras we mean a lattice L of subalgebras whose meet distributes over its join; inf(A,sup(B,C))=sup(inf(A,B),inf(A,C)).
(II) It is very well known and easy to show that every finite commutative algebra can be uniquely decomposed into a direct sum of local algebras. However, who might I accredit this to ? Who first formulated this result ? Or, what early and general theory in commutative algebra is this a direct result of in the literature ?