  An ideal I of a commutative ring R is said to be irreducible if it cannot be
written as the intersection of two larger ideals. A proper ideal I of a ring R
is said to be strongly irreducible if for each ideals J, K of R, J\cap
K\subseteq I implies that J\subset I or K\subset I. In this paper, we introduce
the concepts of 2-irreducible and strongly 2-irreducible ideals which are
generalizations 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 or
I=K\cap L. A proper ideal I of a ring R is called strongly 2-irreducible if for
each ideals J, K and L of R, J\cap K\cap L\subseteq I implies that either J\cap
K\subseteq I or J\cap L\subseteq I or K\cap L\subseteq I.
