On 2-absorbing ideals of non-commutative semirings

Let S be a non-commutative semiring with 1 not equal 0. We study the notions of 2-absorbing and strong 2-absorbing ideals of S and we show that if the semiring is commutative, then these two notions are the same. Also, we give an example to show that in general, these two notions are different. Many properties of (strong) 2-absorbing ideals are proved as a generalization to the results for those over rings. For example, we show that a proper subtractive ideal P of a semiring S is a 2-absorbing ideal of S if and only if whenever I, J and K are left ideals of S such that IJK subset of P, then IJ subset of P or IK subset of P or JK subset of P.