Intermediate Ideals

IntermediateIdeals(I, J): AlgEtQIdl, AlgEtQIdl -> SetIndx[AlgEtQIdl]

Minimal                    : BoolElt                    Default: false
Maximal                    : BoolElt                    Default: false
PrescribedMultiplicatorRing: BoolElt                    Default: false

Given fractional \(S\)-ideals \(J \subset I\), returns all the fractional \(S\)-ideals \(K\) such that \(J \subset K \subset I\).

If Minimal is set true, only the minimal ideals are returned. If Maximal is set true, only the maximal ideals are returned. If PrescribedMultiplicatorRing is set true, only ideals \(K\) with \((K:K) = S\) are returned. The computation is done recursively starting with the minimal or maximal ones.

IntermediateIdeals(I, J, O): AlgEtQIdl, AlgEtQIdl, AlgEtQOrd -> SetIndx[AlgEtQIdl]

PrescribedMultiplicatorRing: BoolElt                    Default: false

Given fractional \(S\)-ideals \(I\) and \(J\) and an order \(O\) such that \(S \subseteq O\), \(J \subseteq I\), and \(O \subseteq (I:I)\), this function returns all the fractional \(S\)-ideals \(K\) such that

• \(J \subseteq K \subseteq I\), and

• \(O \cdot K = I\).

If PrescribedMultiplicatorRing is set true, then the output contains only \(K\) such that \((K:K)=S\). Note that the output may contain \(I\). The output is produced by recursively computing maximal intermediate ideals.

IntermediateIdeals(I, J, N): AlgEtQIdl, AlgEtQIdl, RngIntElt -> SetIndx[AlgEtQIdl]

Given ideals \(J \subset I\) over the same order, and a positive integer \(N\), it returns all the ideals \(K\) such that

• \(J \subset K \subset I\), and

• \([I:K]=N\).

These are computed by recursively searching for maximal submodules.