On 2-Absorbing Primary Ideals in Commutative Semirings

In this paper, we define 2-absorbing and weakly 2-absorbing primary ideals in a commutative semiring S with 1 not equal 0 which are generalization of primary ideals of commutative ring. A proper ideal I of a commutative semiring S is said to be a 2-absorbing primary (weakly 2-absorbing primary) ideal of S if abc is an element of I (0 not equal abc is an element of I) implies ab is an element of I or bc is an element of root I or ac is an element of root I. Some results concerning 2-absorbing primary and weakly 2-absorbing primary ideals are given. It is proved that a subtractive weakly 2-absorbing primary ideal I that is not a 2-absorbing primary ideal satisfies root I = root 0.