The Maximal Orders Which Contain a Given Idealiser Ring

Let C be a commutative Dedekind domain, let S be a maximal C-order in a simple Artinian ring Q, and let K be a proper essential right ideal of S with SK = S. We shall study those maximal C-orders in Q which contain the idealiser ring R of K and the way in which invertible ideals of R act on them by conjugation. In a special case, which includes that in which R is hereditary, we shall show that these maximal orders are in one-to-one correspondence with the ideals of S which contain the bound of K, and we shall give a complete description of the invertible ideals of R.