Abstract
This chapter discusses the structure and classification of hereditary noetherian prime rings. A HNP-ring R is a pseudo-Dedekind ring if R has only a finite number of idempotent ideals, and every nonzero ideal of R contains an invertible ideal. It is known that pseudo-Dedekind rings have a rather uncomplicated ideal theory and that they form a very extensive class of HNP-rings. A HNP-ring R is a Dedekind ring if R has no proper idempotent ideals. In fact, a Dedekind domain is a domain that is also a Dedekind ring. It is observed that Dedekind domains need not be commutative. The chapter presents a structure theorem for arbitrary pseudo-Dedekind rings is presented. It presents all the known structure theoretic results for various classes of pseudo-Dedekind rings and provides new information even in the classical case.
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