On 2-Absorbing Ideals Of Non-Commutative Semirings

Let Formula: see text be a non-commutative semiring with Formula: see text. We study the notions of 2-absorbing and strong 2-absorbing ideals of Formula: see text 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 Formula: see text of a semiring Formula: see text is a 2-absorbing ideal of Formula: see text if and only if whenever Formula: see text, Formula: see text and Formula: see text are left ideals of Formula: see text such that Formula: see text, then Formula: see text or Formula: see text or Formula: see text.