Take the 2-minute tour ×  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 $A_i$.  share|improve this question
You have to be careful when considering the sum A_1 + ... + A_n.
Probably if you are interested in commutative rings then you
should call it a produc