Rings

Rings are algebraic structures consisting of a set equipped with two binary operations: addition and multiplication. In a ring R, the pair (R, +) forms an abelian group, while (R, ×) forms a monoid with a multiplicative identity, typically denoted 1. Multiplication distributes over addition from both sides. Some rings include a multiplicative identity (rings with unity); others do not. If multiplication is commutative, the ring is called a commutative ring. Subsets closed under both operations and containing the zero element are subrings, and subrings containing the unity are unital subrings. An ideal is an additive subgroup closed under multiplication by any ring element, enabling the construction of quotient rings R/I. Ring homomorphisms preserve addition and multiplication, with kernels measuring non-injectivity and images describing the map’s range. Common examples include the integers Z, polynomial rings R[x] over any ring R, and matrix rings Mn(R), which are typically noncommutative for n > 1. Key concepts include zero divisors, integral domains (rings with no zero divisors), and fields (commutative rings in which every nonzero element is a unit). The characteristic of a ring is the smallest positive n such that n·1 = 0, or 0 if no such n exists. Rings underpin many areas of mathematics, providing a unifying language for structures encountered in algebra, number theory, geometry, and physics.

In everyday language, a ring also refers to a circular band worn on a finger. Jewelry rings are made from metals such as gold, platinum, or silver, often alloyed for strength, and come in various karats, sizes, and finishes. Common types include wedding bands, engagement rings, and eternity rings, as well as signet rings. Settings may hold gemstones with prongs, bezels, or channels. Rings symbolize commitment, status, or personal style and are designed with aesthetics and meaning in mind. Manufacturing methods include casting, forging, and fabrication, with sizing and engraving performed after forming.