Narrow Results

Skew-braces have been introduced recently by Guarnieri and Vendramin. The structure group of a non-degenerate solution to the Yang–Baxter equation is a skew-brace, and every skew-brace gives a set-theoretic solution to the Yang–Baxter equation. It is proved that skew-braces arise from near-rings with a distinguished exponential map. For a fixed skew-brace, the corresponding near-rings with exponential form a category. The terminal object is a near-ring of self-maps, while the initial object is a near-ring which gives a complete invariant of the skew-brace. The radicals of split local near-rings with a central residue field $K$ are characterized as $K$-braces with a compatible near-ring structure. Under this correspondence, $K$-braces are radicals of local near-rings with radical square zero.

For near-ring ideal mappings ρ₁ and ρ₂, we investigate radical theoretical properties of and the relationship among the class pairs (ρ₁: ρ₂), and (ℛ_ρ2: ℛ_ρ1). Conditions on ρ₁ and ρ₂ are given for a general class pair to form a radical class of various types. These types include the Plotkin and KA-radical varieties. A number of examples are shown to motivate the suitability of the theory of Hoehnke-radicals over KA-radicals when radical pairs of near-rings are studied. In particular, it is shown that forms a KA-radical class, where denotes the class of completely prime near-rings and the class of 3-prime near-rings. This gives another near-ring generalization of the 2-primal ring concept. The theory of radical pairs are also used to show that in general the class of 3-semiprime near-rings is not the semisimple class of the 3-prime radical.

The flow (or lack thereof) of several kinds of primeness between a zero-symmetric near-ring R and its group near-ring R[G] for certain groups G is discussed. In certain cases, results are contrasted against what happens in the matrix near-ring situation.

Let N be a prime near-ring. We show two main results on the commutativity of N: (1) If there exist k, l ∈ ℕ such that N admits a generalized derivation D satisfying either D([x,y])=x^k [x,y]x^l for all x, y ∈ N or D([x,y])=-x^k [x,y]x^l for all x, y ∈ N, then N is a commutative ring. (2) If there exist k, l ∈ ℕ such that N admits a generalized derivation D satisfying either D(x ◦ y)= x^k (x ◦ y) x^l for all x, y ∈ N or D(x ◦ y)= -x^k (x ◦ y) x^l for all x, y ∈ N, then N is a commutative ring. Moreover, some interesting relations between the prime graph and zero-divisor graph of N are studied.

In this paper, we associate the graph Γ_I(N) to an ideal I of a near-ring N. We exhibit some properties and structure of Γ_I(N). For a commutative ring R, Beck conjectured that both chromatic number and clique number of the zero-divisor graph Γ(R) of R are equal. We prove that Beck's conjecture is true for Γ_I(N). Moreover, we characterize all right permutable near-rings N for which the graph Γ_I(N) is finitely colorable.

This chapter introduces background, history, and recent results on centers and generalized centers of near-rings.

In 1933, P. Jordan proposed a quantum mechanical formalism in which the operators form only a near-algebra. In this regard, the investigation of near-algebras is interesting not only as an axiomatic question but also for physical reasons. A near-algebra is a near-ring which admits a field as a right operator domain. Near-algebras occur more naturally as mappings of linear spaces. The concept Gamma ring (or Γ-ring) (a generalization of a ring) was introduced by Nobusawa [5] and generalized by Barnes [1]. A generalization of both the concepts near-ring and the Γ-ring namely Γ-near-ring was introduced by Satyanarayana [6]. Brown [2], Yamamuro [11], Irish [3] and Srinivas [4;7;8;9] have studied the structure of near-algebras. Γ-near-algebra is a generalization of both the concepts of nearalgebra and Γ-near-ring is introduced by Srinivas et al. [10]. Throughout this lecture, “near-algebra” we mean a “right near-algebra”, and X denotes a field.

Let S =(Q,A,F) be a homomorphic GSA (group-semiautomaton). We obtain: N(S) = {∑ ± f_αi | α_i ∈ A*} = : N, and N(S) is a general distributively generated near-ring with identity. Especially, if S is even linear, then N(S) is an affine near-ring with identity.