Exercise 12, § 3

Let $A = \prod_{i=1}^n A_i$, be the product of a finite family of local rings $A_i$, canonically identified with ideals of A. Let B be a subring of A such that $\mathrm{pr}_i B = A_i$ for $1 \leq i \leq n$. Show that B is the direct composition of at most n local rings and can only be the direct composition of n local rings if $B = A$. (Proceed by induction on n, considering the ideals $a_i = B \cap A_i$ of B. Examine successively the case where $a_i = A_i$ for at least one index i and the case where $a_i \neq A_i$ for all i. Note that, if $a_i \neq A_i$, every maximal ideal of B contains $a_i$, and considering the ring $B/a_i$, conclude that there are at most $n - 1$ distinct maximal ideals in $B$; if then $B$ is not a local ring, reduce it to the case where $a_j = A$, for some $j \neq i$ using the induction hypothesis and Exercise 1 (b.).) Give an example where $n > 1$ and $B$ is a local ring but not isomorphic to any $A_i$ (take the $A_i$ to be algebras over the same field $K$, whose maximal ideals $m$, are of zero square).