Direct Sums of Prime Subsemimodules

Let be a commutative semiring. A semimodule over a semiring is a fully prime semimodule if each proper subsemimodule of is prime. This research aims to investigate the relationship between a direct sum of prime subsemimodules and , , and a fully prime semimodule.

On 2-absorbing ideals of commutative semirings

Let [Formula: see text] be a commutative semiring with unity different than zero. In this paper, we study the concept of [Formula: see text]-absorbing ideals of [Formula: see text] which can be considered as a generalization of prime ideals. Among others, it is shown that the radical of a [Formula: see text]-absorbing ideal is also a [Formula: see text]-absorbing ideal and there are at most [Formula: see text] prime [Formula: see text]-ideals of [Formula: see text] that are minimal over a [Formula: see text]-absorbing ideals.

Divisibility Properties of the Semiring of Ideals of an Integral Domain

Let D be an integral domain, F+(D) (resp., f+(D)) be the set of nonzero (resp., nonzero finitely generated) ideals of D, R1 = f+(D) ∪ {(0)}, and R2 = F+(D) ∪ {(0)}. Then (Ri, ⊕, ⊗) for i = 1, 2 is a commutative semiring with identity under I ⊕ J = I + J and I ⊗ J = IJ for all I, J ∈ Ri. In this paper, among other things, we show that D is a Prüfer domain if and only if every ideal of R1 is a k-ideal if and only if R1 is Gaussian. We also show that D is a Dedekind domain if and only if R2 is a unique factorization semidomain if and only if R2 is a principal ideal semidomain. These results are proved in a more general setting of star operations on D.

Some results about ϕ-primal subsemimodules

Let [Formula: see text] be a commutative semiring with nonzero identity and [Formula: see text] an [Formula: see text]-semimodule. In this paper, we introduce the concept of [Formula: see text]-primal subsemimodule of [Formula: see text] that is a generalization of primal ideal of a commutative ring. Then we give some examples and properties of these subsemimodules. Also, some characterizations of [Formula: see text]-primal subsemimodules are presented.

On B-Ideals in Semirings

In this paper, we introduce the notion of <em>B</em>-ideal in a commutative semiring <em>R</em>. Then 1) A characterization of <em>B</em>-ideals in the Semiring of non-negative integers is obtained. 2) Relation between <em>B</em>-ideals in a semiring <em>R</em> containing a <em>Q</em>-ideal <em>I</em> of <em>R</em> and <em>B</em>-ideals in the quotient semiring <em>R/I_(Q)</em> is obtained. Further study of <em>k</em>-Noetherian semirings is developed. Also <em>B</em>-ideals in polynomial semirings are studied.

Radicals of semirings

It is shown that the classes [Formula: see text] and [Formula: see text] of semirings are radical classes, where [Formula: see text] is the class of subtractive-simple right [Formula: see text]-semimodules and [Formula: see text] is the class of right [Formula: see text]-semimodules isomorphic to [Formula: see text] for some maximal-subtractive right ideal [Formula: see text] of [Formula: see text]. We define the lower Jacobson Bourne radical [Formula: see text] and upper Jacobson Bourne radical [Formula: see text] of [Formula: see text]. For a semiring [Formula: see text], [Formula: see text] holds, where [Formula: see text] is the Jacobson Bourne radical of [Formula: see text]. The radical [Formula: see text] and also coincides with [Formula: see text], if we restrict the class [Formula: see text] to additively cancellative semimodules[Formula: see text] The upper radical [Formula: see text] and [Formula: see text][Formula: see text], if [Formula: see text] is additively cancellative. Further, [Formula: see text], if [Formula: see text] is a commutative semiring with [Formula: see text] The subtractive-primitiveness and subtractive-semiprimitiveness of [Formula: see text] are closely related to the upper radical [Formula: see text] Finally, we show that [Formula: see text]-semisimplicity of semirings are Morita invariant property with some restrictions.

On weakly n-absorbing subtractive ideals in semirings

Let [Formula: see text] be a commutative semiring with identity [Formula: see text]. In this paper weakly [Formula: see text]-absorbing ideal of [Formula: see text] which is a generalization of weakly prime ideal of [Formula: see text] is introduced and some related results are obtained.

Generalization of Semiprime Subsemimodules

The authors introduce the concept of almost semiprime subsemimodules of semimodules over a commutative semiring R. They investigated some basic properties of almost semiprime and weakly semiprime subsemimodules and gave some characterizations of them, especially, for (fnitely generated faithful) multiplication semimodules. They also study the relations among the semiprime, weakly semiprime and almost semiprime subsemimodules of semimodules over semirings.Journal of Bangladesh Academy of Sciences, Vol. 40, No. 2, 187-197, 2016