Let $R$ be a (commutative) ring (with identity). A nonzero idempotent $e\in R$ is called primitive idempotent, whenever it has no decomposition into $a+b$ where $a$ and $b$ are nonzero orthogonal ($ab=0$) idempotents in $R$. I am looking for a characterization for rings with the property that if $e$ is a primitive idempotent of $R$ and $e=r+s$, where $r, s\in R$, then $(e)\subseteq (r)$ or $(e)\subseteq (s)$ where $(x)$ is the ideal generated by $x$ in $R$.


The property you are looking at is that of a local idempotent.

Let $R$ be a commutative ring with $1$. A local idempotent of $R$ is an idempotent $e\in R$ such that $eRe = eR$ is a local ring. Local idempotents are always primitive, but primitive idempotents need not be local (e.g., $1\in\mathbb Z$ is primitive but not local).

Here is the argument that your property is the same as being local.

Assume that $e$ is a local idempotent. Then $R\cong eR\times (1-e)R$, where the factor $eR$ is a local ring. In coordinates, $e = (1,0)$. Suppose that $e=r+s$. Writing this in coordinates yields $(1,0)=(r_1,r_2) + (s_1,s_2)$. Since $r_1, s_1\in eR$, which is local, one of $r_1$ or $s_1$ is a unit in $eR$. Suppose that it is $r_1$. Then $e = (1,0) = (r_1,r_2)*(r_1^{-1},0) \in (r)$, or $(e)\subseteq (r)$. Similarly, if $s_1$ is a unit in $eR$, then $e\in (s)$, or $(e)\subseteq (s)$.

Now suppose that $e$ is not local, so $eR$ is not a local ring. Let $M, N\subseteq eR$ be distinct maximal ideals. Then $M+N=eR$, so there exist $m\in M$ and $n\in N$ such that $m+n=1$ in $eR$. Thus, writing in coordinates in $R \cong eR\times (1-e)R$, we have $(m,0) + (n,0) = (1,0) = e$. I let $r = (m,0)$ and $s=(n,0)$, so $r+s=e$. But $e=(1,0)\notin (r)$, since $(r) \subseteq M\times \{0\}$ and $(1,0)\notin M\times \{0\}$. Similarly $e\notin (s)$.

You have stated ``I am looking for a characterization for rings with the property that if $\ldots$ ETC.'' One (partial?) answer is: those rings whose primitive idempotents are local. I'm not sure how much more you can say than this, but the following paper

W.K. Nicholson.
Trans. Amer. Math. Soc. 207 (1975), 361--373.

proves that primitive idempotents are local in any ``semipotent'' ring. A commutative ring is semipotent if every ideal not contained in the Jacobson radical contains a nonzero idempotent.


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