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Let S be a commutative semiring with unity and M be an unitary left S-semimodule. Let T (M) be the set of all subsemimodules of M and... . [ABSTRACT FROM AUTHOR]

We study some important results of prime and primary subsemimodules. We also prove the primary avoidance theorem for subsemimodules. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the concepts of prime and primary ideals in terms of ɸ-prime and ɸ-primary ideals in a commutative semiring S with non-zero identity. We also introduce the notion of ɸ-subtractive ideals in a commutative semiring to prove some analogous results corresponding to commutative ring theory. [ABSTRACT FROM AUTHOR]

We study 2-absorbing ideals in a commutative semiring S with 1 ≠ 0 and prove some important results analogous to ring theory. More general form of the Prime Avoidance Theorem is also given. We also prove that if I = 〈ha1, a2, ..., ar〉 is a finitely generated ideal of a semiring S and P1, P2,...,Pn are subtractive prime ideals of S such that ⊂ Pi for each 1 ≤ i 6 ≤, then there exist b2,...,br ϵ S such that c = a1 + b2a2 + ... + brar ϵ ⋓ni=1 Pi. [ABSTRACT FROM AUTHOR]

Let S be a commutative semiring with unity and M be an unitary left S-semimodule. Let Φ : T (M) → T (M) ∪ {ø} be a function, where T (M) is the set of all subsemimodules of M. A proper subsemimodule N of M is called a Φ-prime subsemimodule of M if s ∈ S, χ ∈ M, sχ ∈ N \ Φ (N), then either s ∈ (N : M) or χ ∈ N. In this paper, we study the concept of Φ-prime subsemimodule which is a generalization of Φ-prime ideal in a commutative ring and give some characterizations in terms of M-subtractive subsemimodules. [ABSTRACT FROM AUTHOR]

Let S be a commutative semiring and T(S) be the set of all ideals of S. Let Ø: T(S) → T(S) {Ø} be a function. A proper ideal I of a semiring S is called an (n-1, n)-Ø-prime ideal of S if a1a2 ⋯ an ϵ I \ Ø(I), a1, a2,⋯, an ϵ S implies that a1a2 ⋯ ai-1ai+1 ⋯ an ϵ I for some i ϵ {1,2,⋯, n}. In this paper, we prove several results concerning (n-1, n)-Ø-prime ideals in a commutative semiring S with non-zero identity connected with those in commutative ring theory. [ABSTRACT FROM AUTHOR]

In this paper, we define 2-absorbing and weakly 2-absorbing primary ideals in a commutative semiring S with 1 ≠ 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 ∊ I (0 ≠ abc ∊ I) implies ab ∊ I or bc ∊ √I or ac ∊ √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 √I = √0. [ABSTRACT FROM AUTHOR]