In this paper, we introduce the notion of k-derivation, generalized k-derivation and k-reverse derivation on gamma semirings, and we give some commutativity conditions on γ-prime and γ-semiprime ...gamma semirings. Also, we give orthogonality for pairs of k-reverse derivations on gamma semirings.
Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainty. In this paper, we initiate the study of soft semirings by using the soft set ...theory. The notions of soft semirings, soft subsemirings, soft ideals, idealistic soft semirings and soft semiring homomorphisms are introduced, and several related properties are investigated.
Strongly Clean Semiring Das, D.; Kar, S.
National Academy of Sciences, India. Proceedings. Section A. Physical Sciences,
04/2024, Volume:
94, Issue:
2
Journal Article, Conference Proceeding
Peer reviewed
In this paper, we study the concept of strongly clean semiring. Let
S
be a semiring. An element
a
∈
S
is called strongly clean if
a
=
e
+
u
with
e
an idempotent in
S
and
u
a unit in
S
such that
e
u
=
...u
e
. A semiring
S
is said to be strongly clean if every nonzero element of
S
is strongly clean. We mainly study the notion of strongly clean semiring and obtain some important characterizations of strongly clean semiring in connection with exchange semiring, antisimple semiring and inverse semiring.
Clean semiring Kar, S.; Das, D.
Beiträge zur Algebra und Geometrie,
03/2023, Volume:
64, Issue:
1
Journal Article
Peer reviewed
In this paper, we introduce the concept of clean semiring as a generalization of clean ring. A semiring is said to be clean if its every nonzero element can be written as the sum of an idempotent and ...a unit. We study the notion of clean semiring and obtain some important characterizations of clean semiring. We also study the notion of exchange semiring and find out the connenction between clean semiring and exchange semiring.
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.
We study ideal-simple commutative semirings and summarize the results giving their classification, in particular when they are finitely generated. In the principal case of (para)semifields, we then ...consider their minimal number of generators and show that it grows linearly with the depth of an associated rooted forest.
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