On the Semiradical of a Semiring

Abstract

The analytic details in each case will depend upon the nature of the operators A and B. For the case of linear differential operators, the methods given in Bellman and Lehman2 will yield the desired results.

Intuitionistic fuzzy k-ideals of a semiring

“…A semiring is defined by an algebra (S, +, ·) such that (S, +) and (S, ·) are semigroups connected by a(b + c) = ab + ac and (b + c)a = ba + ca for all a, b, c ∈ S. A semiring may have an identity 1, defined by 1 · a = a · 1 = a and a zero 0 (which is an absorbing zero also), defined by 0 + a = a + 0 = a and a · 0 = 0 · a = 0 for all a ∈ S(see [7]). A subset J ( = ∅) of a semiring S is called a left ideal of S, if a + b ∈ J, sa ∈ J for all a, b ∈ J and all s ∈ S. Right ideal is defined dually and a two sided ideal or simply an ideal is both a left and a right ideal(see [7]).…”

“…A subset J ( = ∅) of a semiring S is called a left ideal of S, if a + b ∈ J, sa ∈ J for all a, b ∈ J and all s ∈ S. Right ideal is defined dually and a two sided ideal or simply an ideal is both a left and a right ideal(see [7]). Definition 3.1 [15].…”

“…Right k-ideal is defined dually, and two sided k-ideal or simply a k-ideal is both a left and a right k-ideal (See [7]). Definition 3.2 Let A be a nonempty intuitionistic fuzzy set in a semiring S satisfying the following conditions : for any x, y ∈ S,…”

On maximal 𝑘-ideals of semirings

“…Using only commutativity of addition, the following concepts and statements, essentially due to [1,2,4] Proof. Assume that 5 satisfies (C).…”

“…If also (S, •) is commutative, S is called a commutative semiring. Moreover, to avoid trivial exceptions, each semiring S is assumed to have at least two elements.Using only commutativity of addition, the following concepts and statements, essentially due to [1,2,4], are well known. For each ideal A of a semiring S…”