In what follows, a ring will denote a commutative ring with unity.
I was doing some problems from a Commutative Algebra class and I came across the following problem/idea. It is a common practice to characterize some ideal $I$ of a ring $R$ in terms of a property of the quotient ring $R/I$. We can summarize some of these characterizations in the following result:
Proposition.Let $I$ be an ideal of a ring $R$. Then:
1) $I$ is a prime ideal if and only if $R/I$ is an integral domain;
2) $I$ is a maximal ideal if and only if $R/I$ is a field;
3) $I$ is a radical ideal if and only if $R/I$ is reduced (i.e., has no non-trivial nilpotents).
The problem I was solving was about finding non-trivial ($\neq 0,1$) idempotents in the ring $\mathbb{Z}/n\mathbb{Z}$. Although I solved the problem, I started thinking about rings with no non-trivial idempotents, which I discovered are called connected rings, due to the following beautiful result:
Proposition.A ring $R$ is connected if and only if the spectrum $\mathrm{Spec}(R)$ with the Zariski topology is connected.
Then, the following question came to my mind:
Question.What types of ideals $I \subseteq R$ give rise to a connected quotient $R/I$?
So far, we know that $R/I$ is connected if and only if, given $\bar{x} \in R/I$,
$$ [ \:\: \bar{x}^2=\bar{x} \:\: \Longrightarrow \:\: \bar{x}=\bar{0} \:\: \text{or} \:\: \bar{x}=\bar{1} \:\: ] \Longleftrightarrow [ \:\: x(x-1)\in I \Longrightarrow \:\: x \in I \:\: \text{or} \:\: x-1\in I \:\: ], $$
which suggests the following definition:
Definition. A proper ideal $I \subseteq R$ is called connected (tentative name) if for a given element $x \in R$ such that $x(x-1) \in I$, we have $x \in I$ or $x-1 \in I$.
My (soft) questions are the following: Is there an existing name for these types of ideals? What properties do these ideals have (apart from the obvious, such as any prime ideal being of this type)? Is there a relation between these ideals and radical ideals?
Thank you in advance!