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An ideal I of a commutative ring R is said to be irreducible if it cannot bewritten as the intersection of two larger ideals. A proper ideal I of a ring Ris said to be strongly irreducible if for each ideals J, K of R, J\capK\subseteq I implies that J\subset I or K\subset I. In this paper, we introducethe concepts of 2-irreducible and strongly 2-irreducible ideals which aregeneralizations of irreducible and strongly irreducible ideals, respectively.We say that a proper ideal I of a ring R is 2-irreducible if for each ideals J,K and L of R, I= J\cap K\cap L implies that either I=J\cap K or I=J\cap L orI=K\cap L. A proper ideal I of a ring R is called strongly 2-irreducible if foreach ideals J, K and L of R, J\cap K\cap L\subseteq I implies that either J\capK\subseteq I or J\cap L\subseteq I or K\cap L\subseteq I.

2-irreducible and strongly 2-irreducible ideals of commutative rings