Ring

A ring is a term with multiple meanings in mathematics, everyday objects, and culture. In algebra, a ring is a set equipped with two binary operations: addition and multiplication. The structure must be closed under these operations, be associative for both, contain an additive identity (0), and provide additive inverses for every element. Multiplication distributes over addition. If a ring has a multiplicative identity (1) it is called a ring with unity; if multiplication is commutative, it's a commutative ring. Ring theory studies these objects, their ideals, homomorphisms, and factor structures. Common examples include the integers, polynomial rings, and rings of matrices.

In jewelry, a ring is a circular band worn on a finger. Rings range from plain metal bands to elaborate pieces set with gemstones. They symbolize relationships such as engagement and marriage, personal milestones, or fashion. Materials include gold, silver, platinum, and alternative metals, with finishes like polishing, engraving, or enamel. Designers may create rings with different profiles, thicknesses, and widths, and rings may be worn singly or stacked.

Other uses include the boxing or wrestling ring, a designated area surrounded by ropes where matches occur. A ring also denotes a sound signal, as in a telephone ringing to indicate an incoming call; the verb to ring means to cause such a sound or to call by telephone. In physics and chemistry, rings appear as cyclic structures; for example, aromatic rings such as benzene are stable carbon rings that underpin many compounds.