| Summary: | The paper is devoted to the investigation of the notion of a differentially prime ideal of a differential commutative semiring (i. e. a semiring equipped with a derivation), and its interrelation with the notions of a quasi-prime ideal and a primary ideal. The notion of a semiring derivation is traditionally defined as an additive map satisfying the Leibnitz rule, i. e. a map δ: R → R is called a derivation on R if δ (a + b)= δ (a) + δ (b) and δ (ab) = δ (a)b + aδ (b) for any a, b ∈ R.
A differential ideal P of R is called a differentially prime ideal if for any a, b ∈ R, k ∈ ℕ0, ab(k) ∈ P follows a ∈ P or b ∈ P. It is proved that an ideal P of a semiring R is prime if and only if for any ideals I and J of R the inclusion IJ ⊆ P follows I ⊆ P or J ⊆ P. A quasi-prime ideal is a differential ideal of a semiring which is maximal among those ideals disjoint from some multiplicatively closed subset of a semiring.
In this paper we investigate some properties of such differentially prime ideals, in particular in case of differential Noetherian semirings. The paper consists of two main parts. The first part of the paper is devoted to establishing some properties of differentially prime ideals and gives some examples of such ideals. In the second part, the author investigates the connection existing between quasi-prime ideals, primary ideals and differentially prime ideals in differential Noetherian semirings. It is established that in a differential Noetherian semiring R a differential ideal I of R is differentially prime if and only if I is a quasi-prime ideal.
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