Amalgamated duplication of some special rings along an ideal

Let be a commutative Noetherian ring and let I be a proper ideal of . D’Anna and Fontana in [6] introduced a new construction of ring, named amalgamated duplication of along I. In this paper by considering the ring homomorphism , it is shown that if , then , also it is proved that if , then there exists such that . Using this result it is shown that if is generically Cohen-Macaulay (resp. generically Gorenstein) and is generically maximal Cohen-Macaulay (resp. a generically canonical module), then is generically Cohen-Macaulay (resp. generically Gorenstein). We also defined the notion of generically quasi-Gorenstein ring and we investigate when is generically quasi-Gorenstein. In addition, it is shown that is approximately Cohen-Macaulay if and only if R is approximately Cohen-Macaulay, provided some special conditions. Finally it is shown that if R is approximately Gorenstein, then is approximately Gorenstein.