ABSTRACT. This paper investigates ideal-theoretic as well as homological extensions of the Prüfer domain concept to commutative rings with zero divisors in an amalgamated duplication of a ring along an ideal. The new results both compare and contrast with recent results on trivial ring extensions (and pullbacks) as well as yield original families of examples issued from amalgamated duplications subject to various Prüfer conditions.

Let f : A → B be a ring homomorphism and let J be an ideal of B. In this paper, we give a characterization for the amalgamation of A with B along J with respect to f (denoted by R f J) to be (uniquely) clean.

In this paper, we give a characterization for the amalgamated duplication of a ring R along an ideal I, denoted by R ⋈ I, to be self-injective.

Let A and B be two rings, let J be an ideal of B and let f : A → B be a ring homomorphism. In this paper, we study when the amalgamation of A with B along J with respect to f is a φ-ring. Hence, we study two different chain conditions over this structure. Namely, the nonnil-Noetherian condition and the Noetherian spectrum condition.

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