Near-ring

Key takeaways

Near-ring belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Near-ring to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Near-ring from memory before moving on to harder problems.

Reference excerpt

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.

Definition A set N together with two binary operations + (called addition) and ⋅ (called multiplication) is called a (right) near-ring if:

N is a group (not necessarily abelian) under addition; multiplication is associative (so N is a semigroup under multiplication); and multiplication on the right distributes over addition: for any x, y, z in N, it holds that (x + y)⋅z = (x⋅z) + (y⋅z). 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 uses right near-rings, while the book of Clay uses left near-rings. An immediate consequence of this one-sided distributive law is that it is true that 0⋅x = 0 but it is not necessarily true that x⋅0 = 0 for any x in N. Another immediate consequence is that (−x)⋅y = −(x⋅y) for any x, y in N, but it is not necessary that x⋅(−y) = −(x⋅y). A near-ring is a rng 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.

Mappings from a group to itself Let G be a group, written additively but not necessarily abelian, and let M(G) be the set {f | f : G → G} of all functions from G to G. An addition operation can be defined on M(G): given f, g in M(G), then the mapping f + g from G to G is given by (f + g)(x) = f(x) + g(x) for all x in G. Then (M(G), +) is also a group, which is abelian if and only if G is abelian. Taking the composition of mappings as the product ⋅, M(G) becomes a near-ring. The 0 element of the near-ring M(G) is the zero map, i.e., the mapping which takes every element of G to the identity element of G. The additive inverse −f of f in M(G) coincides with the natural pointwise definition, that is, (−f)(x) = −(f(x)) for all x in G. If G has at least two elements, then M(G) is not a ring, even if G is abelian. (Consider a constant mapping g from G to a fixed element g ≠ 0 of G; then g⋅0 = g ≠ 0.) However, there is a subset E(G) of M(G) consisting of all group endomorphisms of G, that is, all maps f : G → G such that f(x + y) = f(x) + f(y) for all x, y in G. If (G, +) is abelian, both near-ring operations on M(G) are closed on E(G), and (E(G), +, ⋅) is a ring. If (G, +) is nonabelian, E(G) is generally not closed under the near-ring operations; but the closure of E(G) under the near-ring operations is a near-ring. Many subsets of M(G) form interesting and useful near-rings. For example:

The mappings for which f(0) = 0. The constant mappings, i.e., those that map every element of the group to one fixed element. The set of maps generated by addition and negation from the endomorphisms of the group (the "additive closure" of the set of endomorphisms). If G is abelian then the set of endomorphisms is already additively closed, so that the additive closure is just the set of endomorphisms of G, and it forms not just a near-ring, but a ring. Further examples occur if the group has further structure, for example:

The continuous mappings in a topological group. The polynomial functions on a ring with identity under addition and polynomial composition. The affine maps in a vector space. Every near-ring is isomorphic to a subnear-ring of M(G) for some G.

Applications 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.e., those that are neither rings nor near-fields. The best known is to balanced incomplete block designs 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. James R. Clay and others have extended these ideas to more general geometrical constructions.

See also Near-field (mathematics) Semiring Near-semiring

References

Celestina Cotti Ferrero; Giovanni Ferrero (2002). Nearrings: Some Developments Linked to Semigroups and Groups. Kluwer Academic Publishers. ISBN 978-1-4613-0267-4.

External links The Near Ring Main Page at the Johannes Kepler Universität Linz

Worked examples

Example 1 — a first encounter with Near-ring

Start with the simplest possible case. Write down what Near-ring claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Near-ring before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Near-ring ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Near-ring

In research: Near-ring appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Near-ring in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Near-ring is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic structures, Ring theory, so understanding it makes those chapters shorter.
In everyday life: Look for Near-ring outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Near-ring in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Near-ring means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Near-ring out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Near-ring in simple terms?

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.

Why does Near-ring matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Near-ring?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Near-ring.