Suppose $A$ is a central simple algebra over a field $F$, $\mathcal{O}_F$ is an integrally closed subring of $F$ and suppose $\mathcal{O}_F$ is noetherian. Then an order of $A$ is a $\mathcal{O}_F$ lattice of $A$, stable under the algebra operations. In this case, the maximal order always exists( assume dim$_F A$ is coprime to the char of F). My questions:

Are there some nontrivial criterion for an order being maximal? And how to construct a maximal order starting from a given order? (I only know the existence from some ACC argument)

In fact, what I really concern is "why we could always find a maximal order stable under a given involution on $A$" (It is wrong from Florian's example).

So is there some similar statement (under some condition) which is right? (I kind of heard of it before...)

Thanks.

Suppose $A$ is a central simple algebra over a field $F$, $\mathcal{O}_F$ is an integrally closed subring of $F$ and suppose $\mathcal{O}_F$ is noetherian. Then an order of $A$ is a $\mathcal{O}_F$ lattice of $A$, stable under the algebra operations. In this case, the maximal order always exists( assume dim$_F A$ is coprime to the char of F). My questions:

Are there some nontrivial criterion for an order being maximal? And how to construct a maximal order starting from a given order? (I only know the existence from some ACC argument)

In fact, what I really concern is "why we could always find a maximal order stable under a given involution on $A$" (It is wrong from Eisele's example).

So is there some similar statement (under some condition) which is right? (I kind of heard of it before...)

Thanks.

Suppose $A$ is a central simple algebra over a field $F$, $\mathcal{O}_F$ is an integrally closed subring of $F$ and suppose $\mathcal{O}_F$ is noetherian. Then an order of $A$ is a $\mathcal{O}_F$ lattice of $A$, stable under the algebra operations. In this case, the maximal order always exists. My questions:

Are there some nontrivial criterion for an order being maximal? And how to construct a maximal order starting from a given order? (I only know the existence from some ACC argument)

In fact, what I really concern is why we could always find a maximal order stable under a given involution on $A$.

Any hint is welcome! Thanks.

Suppose $A$ is a central simple algebra over a field $F$, $\mathcal{O}_F$ is an integrally closed subring of $F$ and suppose $\mathcal{O}_F$ is noetherian. Then an order of $A$ is a $\mathcal{O}_F$ lattice of $A$, stable under the algebra operations. In this case, the maximal order always exists( assume dim$_F A$ is coprime to the char of F). My questions:

Are there some nontrivial criterion for an order being maximal? And how to construct a maximal order starting from a given order? (I only know the existence from some ACC argument)

In fact, what I really concern is "why we could always find a maximal order stable under a given involution on $A$" (It is wrong from Florian's example).

So is there some similar statement (under some condition) which is right? (I kind of heard of it before...)

Thanks.