ADS-property for \(H\)-radicals (Q1311104)

We consider conditions under which the ADS property (for a radical \(\rho\), algebra \(A\), and ideal \(B\vartriangleleft A\), we have \(\rho(B) \vartriangleleft A\)) is fulfilled for the radicals of \(H\)-module algebras over an arbitrary bi-algebra \(H\) with comultiplication \(\Delta: H\to H_ k\otimes H\) and co-unit \(\varepsilon: H\to k\). Here \(k\) denotes some fixed field; all the algebras considered are \(k\)-algebras.