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Homesubdirectly irreducible ring

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# subdirectly irreducible ring

A ring $R$ is said to be subdirectly irreducible if every subdirect product representation 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 $\epsilon:R\to\prod R_{i}$ given by $\epsilon(r)(i)=r+I_{i}$ is injective. If $\epsilon(r)=0$, 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, $\bigcap I_{i}\neq 0$.

$(\Leftarrow)$. Suppose next that $\bigcap I_{i}\neq 0$. 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, $J\neq 0$. 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 $I\neq 0$ by subdirect irreducibility. Pick $0\neq s\in I$. 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 $0\neq r\in R$, then $R=I\subseteq rR$, so $1=pr$ for some $p\in R$, which means $R$ is a field.

## Mathematics Subject Classification

16D70*no label found*

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