Quasi-Frobenius Amalgamated Algebras

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Let $$f:A\rightarrow B$$ be a homomorphism of commutative rings and let J be an ideal of B. The amalgamation of A with B along J with respect to f is the subring of $$A\times B$$ given by $$A\bowtie ^fJ=\{(a,f(a)+j)\mid a\in A, \, j\in J\}$$ . In this paper, we give some characterizations for the amalgamation construction to be a quasi-Frobenius ring.