5

Is it true that any prime ideal of a finite direct product ring $R=\prod_{i=1}^nR_i$ is of the form $P=\prod_{i=1}^nI_i$, where $I_j$ is a prime ideal of $R_j$ for some $j$ and $I_i=R_i$, for $i\neq j$?

Any ideal of the form above is prime in $R$. Indeed, if $X=\prod_{i=1}^nA_i$ and $Y=\prod_{i=1}^nB_i$ are ideals of $R$ (where $A_i$'s and $B_i$'s are ideals of $R_i$) such that $XY\subseteq P$, then either $A_j$ or $B_j$ is a subset of a prime ideal $I_j$, for some $j$. Hence either $X$ or $Y$ is a subset of $P$.

Any help/suggestion would be appreciated!

user26857
  • 52,094
karparvar
  • 5,730
  • 1
  • 14
  • 36
  • Yes. In your ring there are $n$ central orthogonal idempotents $e_1,\ldots, e_n$ adding up to $1$, and at most one of these can be outside of $P$, because if $e_i, e_j \notin P$ with $i\neq j$ then $0 = e_i.e_j \in P$ gives a contradiction. On the other hand, if all of the $e_i$ lie in $P$ then $1 = e_1+\ldots+e_n \in P$ which is impossible since prime ideals are proper. So, exactly one $e_j$ lies outside $P$, and $1 - e_j$, which equals the sum of all the other ones, lies in $P$. It follows that $P$ has the form you want. – Konstantin Ardakov Oct 09 '17 at 09:59

1 Answers1

5

If I am not mistaken, you've proven ideals of that form are prime, and now you are headed for the converse. Let $I$ be a prime ideal of $R$.

I assume you already know the ideal structure of finite products of rings, so you can see why the ideal must be of the form $I=\prod I_i$ for some ideals $I_i\lhd R_i$. Then $R/I=(\prod R_i)/(\prod I_i)\cong\prod R_i/I_i$.

Now, in order for this product to be a prime ring (which is exactly what a quotient by a prime ideal gives you) it is necessary for all factors to be zero except one, and the one that is nonzero must also be a prime ring.

The first condition implies there is a $j$ such that $R_k=I_k$ for all $k\neq j$, and the second condtion implies $I_j$ is a prime ideal of $R_j$.

A quick note about KonstantinArdakov's comment-solution: it's fine, except for a small danger of readers misunderstanding part of the logic.

As we know, the general definition of prime ideals says that $AB\subseteq P$ implies $A\subseteq P$ or $B\subseteq P$ for any two ideals $A,B$ of $R$. That is, it's the ideal-wise version of the element-wise condition for primality. So something needs to be said about why $e_ie_j=0$ implies one of the two is in $P$.

Now, given that at least one of $a$ and $b$ is central, we can conclude that $(a)(b)=\{0\}\subseteq P$, and it follows that $a\in P$ or $b\in P$, so everything works as hoped.

rschwieb
  • 153,510