Given a commutative Artin algebra $A$ over an algebraically closed field $k$ one has a decomposition $A=A\_1\oplus\ldots\oplus A\_n$ into local Artin subalgebras, see for example *Atiyah-McDonald, Introduction To Commutative Algebra, Theorem 8.7*. The subalgebras $A\_i$ are uniquely determined up to the isomorphism.


The question is as follows. Are the inclusions $A\_i\subset A$ uniquely determined as well?
They should be, but I cannot find an accurate proof.


UPD: So, the inclusions are not necessarily unique. But may there exist an infinite number of inclusions? Or the number of inclusions is necessarily finite?


Motivation: If there is a finite number of ways for the embedding $A\_i\to A$ then the connected group of unity $(Aut A)^{\circ}$ of the automorphism group of algebra $A$ stabilizes the subalgebra $A\_i$.



local Artin algebras
Yes, the decomposition is unique. The uniqueness of inclusions is a moot point because rings may have nontrivial endomorphisms.


The proof goes like this: consider decompositions of $1$ into the sums of orthogonal idempotents $1=\sum\_i p\_i$. Orthogonality means that $p\_ip\_j=0$ whenever $i\neq j$. From general nonsense (commutativity will be needed) you can find unique maximal decomposition and then $A\_i = p\_iA$.


