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Let $B$ be a commutative unitary reduced ring and let $A$ be a subring of it. Let $e$ be an idempotent of $B$. Then we have a natural surjective ring homomorphism $A\rightarrow Ae$ defined by $a\mapsto ae$. $Ae$ is just the ring with $e$ as unity and multiplication, addition induced from $B$ (i.e. $ae\cdot be = abe$ and $ae+be = (a+b)e$).

The question is.. How much do we know about Spec $Ae$ ?

Edit: Well I realize that the prime ideals of $Ae$ should be of the form $\mathfrak{p}e$ for some prime ideal $\mathfrak{p} \in$ Spec $A$. But how much do we know about the topology of Spec $Ae$?

Edit: Apparently I won't know much about Spec $Ae$ unless I know more about $B$. In my specific problem, I wouldn't mind $B$ to be a von Neumann regular ring of fractions of $A$ (i.e. for all $b\in B\backslash${0} there is an $a$ $\in A$ such that $ab$ $\in A\backslash${0}) with extremally disconnected spectrum. I just didn't wanted to add this extra condition to avoid confusion.

Let $B$ be a commutative unitary reduced ring and let $A$ be a subring of it. Let $e$ be an idempotent of $B$. Then we have a natural surjective ring homomorphism $A\rightarrow Ae$ defined by $a\mapsto ae$. $Ae$ is just the ring with $e$ as unity and multiplication, addition induced from $B$ (i.e. $ae\cdot be = abe$ and $ae+be = (a+b)e$).

The question is.. How much do we know about Spec $Ae$ ?

Edit: Well I realize that the prime ideals of $Ae$ should be of the form $\mathfrak{p}e$ for some prime ideal $\mathfrak{p} \in$ Spec $A$. But how much do we know about the topology of Spec $Ae$?

Edit: Apparently I won't know much about Spec $Ae$ unless I know more about $B$. In my specific problem, I wouldn't mind $B$ to be a von Neumann regular (i.e. zero dimensional) ring of fractions of $A$ (i.e. for all $b\in B\backslash${0} there is an $a$ $\in A$ such that $ab$ $\in A\backslash${0}) with extremally disconnected spectrum (i.e. closure of any open set in Spec $B$ is open). I just didn't wanted to add this extra condition to avoid confusion. My more specific question was whether the ring $A[e]$ could have an extremally disconnected minimal prime spectrum if the minimal prime spectrum of $A$ werent exd, for that it suffices for me to know this for $Ae$.

Let $B$ be a commutative unitary reduced ring and let $A$ be a subring of it. Let $e$ be an idempotent of $B$. Then we have a natural surjective ring homomorphism $A\rightarrow Ae$ defined by $a\mapsto ae$. $Ae$ is just the ring with $e$ as unity and multiplication, addition induced from $B$ (i.e. $ae\cdot be = abe$ and $ae+be = (a+b)e$).

The question is.. How much do we know about Spec $Ae$ ?

Edit: Well I realize that the prime ideals of $Ae$ should be of the form $\mathfrak{p}e$ for some prime ideal $\mathfrak{p} \in$ Spec $A$. But how much do we know about the topology of Spec $Ae$?

Edit: Apparently I won't know much about Spec $Ae$ unless I know more about $B$. In my specific problem, I wouldn't mind $B$ to be a von Neumann regular quotient ring of $A$ (i.e. for all $b\in B\backslash${0} there is an $a\in A$ such that $ab\in A\backslash${0}) with extremally disconnected spectrum. I just didn't wanted to add this extra condition to avoid confusion.

Let $B$ be a commutative unitary reduced ring and let $A$ be a subring of it. Let $e$ be an idempotent of $B$. Then we have a natural surjective ring homomorphism $A\rightarrow Ae$ defined by $a\mapsto ae$. $Ae$ is just the ring with $e$ as unity and multiplication, addition induced from $B$ (i.e. $ae\cdot be = abe$ and $ae+be = (a+b)e$).

The question is.. How much do we know about Spec $Ae$ ?

Edit: Well I realize that the prime ideals of $Ae$ should be of the form $\mathfrak{p}e$ for some prime ideal $\mathfrak{p} \in$ Spec $A$. But how much do we know about the topology of Spec $Ae$?

Edit: Apparently I won't know much about Spec $Ae$ unless I know more about $B$. In my specific problem, I wouldn't mind $B$ to be a von Neumann regular ring of fractions of $A$ (i.e. for all $b\in B\backslash${0} there is an $a$ $\in A$ such that $ab$ $\in A\backslash${0}) with extremally disconnected spectrum. I just didn't wanted to add this extra condition to avoid confusion.