subdirectly irreducible ring

A ring $R$ is said to be subdirectly irreducible if every subdirect product of $R$ is trivial.

Equivalently, a ring $R$ is subdirectly irreducible iff the intersection of all non-zero ideals of $R$ is non-zero.

Proof.

Let $\{I_i\}$ be the set of all non-zero ideals of $R$.

$(\Rightarrow )$. Suppose first that $R$ is subdirectly irreducible. If $\bigcap I_i=0$, then $R$ is a subdirect product of $R_i:=R/I_i$, for $ϵ:R\to \prod R_i$ given by $ϵ(r)(i)=r+I_i$ is injective. If $ϵ(r)=0$, then $r\in I_i$ for all $i$, or $r\in \bigcap I_i=0$, or $r=0$. But then $R\to \prod R_i\to R_i$ given by $r\mapsto r+I_i$ is not an isomorphism for any $i$, contradicting the fact that $R$ is subdirectly irreducible. Therefore, $\bigcap I_i\ne 0$.

$(\Leftarrow )$. Suppose next that $\bigcap I_i\ne 0$. Let $R$ be a subdirect product of some $R_i$, and let $J_i:=\ker (R\to \prod R_i\to R_i)$. Each $J_i$ is an ideal of $R$. Let $J=\bigcap J_i$. If $R\to \prod R_i\to R_i$ is not an isomorphism (therefore not injective), $J_i$ is non-zero. This means that if $R$ is not subdirectly irreducible, $J\ne 0$. But $J\subseteq \ker (R\to \prod R_i)$, contradicting the subdirect irreducibility of $R$. As a result, some $J_i=0$, or $R\to \prod R_i\to R_i$ is an isomorphism. ∎

As an application of the above equivalence, we have that a simple ring is subdirectly irreducible. In addition, a commutative subdirectly irreducible reduced ring is a field. To see this, let $\{I_i\}$ be the set of all non-zero ideals of a commutative subdirectly irreducible reduced ring $R$, and let $I=\bigcap I_i$. So $I\ne 0$ by subdirect irreducibility. Pick $0\ne s\in I$. Then $s^{2}R\subseteq sR\subseteq I$. So $s^{2}R=sR$ since $I$ is minimal. This means $s=s^{2}t$, or $1=st\in sR=I$, which means $I=R$. Now, let any $0\ne r\in R$, then $R=I\subseteq rR$, so $1=pr$ for some $p\in R$, which means $R$ is a field.

Title	subdirectly irreducible ring
Canonical name	SubdirectlyIrreducibleRing
Date of creation	2013-03-22 14:19:13
Last modified on	2013-03-22 14:19:13
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	10
Author	CWoo (3771)
Entry type	Definition
Classification	msc 16D70