# Near Rings The Theory And Its Applications Pdf

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## Near-rings : the theory and its applications

In mathematics , a near-ring also near ring or nearring is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups.

Similarly, it is possible to define a left near-ring by replacing the right distributive law by the corresponding left distributive law. Both right and left near-rings occur in the literature; for instance, the book of Pilz [2] uses right near-rings, while the book of Clay [3] uses left near-rings. A near-ring is a ring not necessarily with unity if and only if addition is commutative and multiplication is also distributive over addition on the left.

If the near-ring has a multiplicative identity, then distributivity on both sides is sufficient, and commutativity of addition follows automatically. The 0 element of the near-ring M G is the zero map , i. If G has at least 2 elements, M G is not a ring, even if G is abelian.

Many subsets of M G form interesting and useful near-rings. For example: [1]. Every near-ring is isomorphic to a subnear-ring of M G for some G. Many applications involve the subclass of near-rings known as near-fields ; for these see the article on near-fields. There are various applications of proper near-rings, i. The best known is to balanced incomplete block designs [2] using planar near-rings. These are a way to obtain difference families using the orbits of a fixed point free automorphism group of a group.

Clay and others have extended these ideas to more general geometrical constructions. From Wikipedia, the free encyclopedia. Algebraic structures Group -like. Ring -like. Lattice -like. Module -like. Module Group with operators Vector space.

Algebra -like. Clay, "Nearrings: Geneses and applications", Oxford, Categories : Algebraic structures Ring theory. Namespaces Article Talk. Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file.

Download as PDF Printable version. Module -like Module Group with operators Vector space Linear algebra.

## Near-semiring

In mathematics , a near-ring also near ring or nearring is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups. Similarly, it is possible to define a left near-ring by replacing the right distributive law by the corresponding left distributive law. Both right and left near-rings occur in the literature; for instance, the book of Pilz [2] uses right near-rings, while the book of Clay [3] uses left near-rings. A near-ring is a ring not necessarily with unity if and only if addition is commutative and multiplication is also distributive over addition on the left.

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This volume showcases mostly the contributions presented at the International Conference in Algebra and Its Applications held at the Aligarh Muslim University, Aligarh, India during November , Refereed by renowned experts in the field, this wide-ranging collection of works presents the state of the art in the field of algebra and its applications covering topics such as derivations in rings, category theory, Baer module theory, coding theory, graph theory, semi-group theory, HNP rings, Leavitt path algebras, generalized matrix algebras, Nakayama conjecture, near ring theory and lattice theory. All of the contributing authors are leading international academicians and researchers in their respective fields. A collection of up-to-date research contributions in algebra and its applications. Refereed by renowned experts in the field. Covers wide-ranging topics such as rings, modules, and algebras.

## Algebra and Its Applications

Pilz, G. Groenewald, N. Algebra, 19 10 , pp. Abbass, H. Journal of Kerbala University, 10 3 , pp.

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