Ideals In Near-rings Research Articles

Minimal ideals in near-rings and localized distributivity conditions

AbstractLetB, S, andTbe subsets of a (left) near-ringRwithBandTnonempty. We sayBis (S, T)-distributiveifs(b1+b2)t=sb1t+sb2t, for eachs∈S,b1,2∈B,t∈T. Basic properties for this type of ‘localized distributivity’ condition are developed, examples are given, and applications are made in determining the structure of minimal ideals.Theorem. IfIis a minimal ideal ofRandIkis (Im,In)-distributive for somek,n≧ 1,m≧ 0, then eitherI2= 0 orIis a simple, nonnilpotent ring with every element ofIdistributive inR. Theorem. LetRkbe (Rm,Rn)-distributive, for somek,n≧ 1, m ≧ 0; ifRis semiprime or is a subdirect product of simple near-rings, thenRis a ring. Connections are established with near-rings which satisfy a permutation identity and with weakly distributive near-rings. IfR→A→ 0 is an exact sequence of near-rings, then conditions onAare given which will impose conditions on the minimal ideals ofR.

Minimal ideals in near-rings

Minimal ideals in near-rings

Maximal left ideals in near-rings of continuous functions on disconnected groups

Let N(G) denote the near-ring of all continuous selfmaps of a topological group G. In this paper we determine the maximal left ideals of N(G) for certain classes of disconnected groups G.

Different prime ideals in near-rings

Different prime ideals in near-rings

Distributor ideals in near-rings of polynomials

Distributor ideals in near-rings of polynomials

A generalization of prime ideals in near-rings

(1984). A generalization of prime ideals in near-rings. Communications in Algebra: Vol. 12, No. 15, pp. 1835-1853.

A characterization of semi-prime ideals in near-rings

AbstractIt is well-known that in any near-ring, any intersection of prime ideals is a semi-prime ideal. The aim of this note is to prove that any ideal is a prime ideal if and only if it is equal to its prime radical. As a consequence of this we have any semi-prime ideal I in a near-ring N is the intersection of minimal prime ideals of I in N and that I is the intersection of all prime ideals containing I.

A Characterization of semiprime ideals in near-rings

AbstractIt is well known that in any near-ring, any intersection of prime ideals is a semiprime ideal. The aim of this paper is to prove that any semiprime ideal I in a near-ring N is the intersection of all minimal prime ideals of I in N. As a consequence of this we have any seimprime ideal I is the intersectionof all prime ideals containing I.

Near-rings of compatible functions

SummaryIn this paper we study near-rings of functions on Ω-groups which are compatible with all congruence relations. Polynomial functions, for instance, are of this type. We employ the structure theory for near-rings to get results for the theory of compatible and polynomial functions (affine completeness, etc.). For notations and results concerning near-rings see e.g. (10). However, we review briefly some terminology from there. (N, +,.) is a near-ring if (N, +) is a group and . is associative and right distributive over +. For instance,M(A): = (AA, +, °) is a near-ring for any group (A, +) (° is composition). IfNis a near-ring thenN0: = {n∈N/n0 = 0}. A group (Γ, +) is anN-group (we writeNΓ) if a “product”nyis defined with (n+n‛)γ =nγ +n‛γ and (nn‛)γ =n(n‛γ). Ideals of near-rings andN-groups are kernels of (N-) homomorphisms. If Γ is a vector-space,Maff(Γ) is the near-ring of all affine transformations on Γ.Nis 2-primitive onNΓ ifNΓ is non-trivial, faithful and without properN-subgroups. The (2-) radical and (2-) semisimplicity are defined similarly to the ring case.

Remark on the preceding paper: “Ideals in near rings of polynomials over a field”

Remark on the preceding paper: “Ideals in near rings of polynomials over a field”

Minimal Ideals of Near-Rings with Minimal Condition

Minimal Ideals of Near-Rings with Minimal Condition

One-sided ideals in near-rings of transformations

Let (G, +) be an arbitrary group and let T0(G) = {f∈ Map(G, G): 0f = 0}; the system composed of T0(G) and the operations of pointwise addition and composition of functions form a (left) near-ring. Berman and Silverman, in their investigation of near-rings of transformations [3], found that for every group G the associated near-ring of transformations T0(G) has no proper ideals. In the present paper left and right ideals of T0(G) are considered.