Where does ring collide?

A set equipped with two operations, addition and multiplication, that obey a fixed list of axioms: the elements form an abelian group under addition, multiplication is associative, and multiplication distributes over addition. The referent is an abstract structure, not any particular collection of numbers; the integers are one ring, but so are polynomials, square matrices, and functions. The idea was distilled in the late nineteenth and early twentieth centuries from Dedekind's work on ideals and Emmy Noether's axiomatic treatment, and it now anchors abstract algebra alongside groups and fields. What marks this meaning is the pair of operations and their axioms. A ring here is defined by what you can do to its elements, not by any loop or circle; the word itself was borrowed from Hilbert's Zahlring and names no shape at all.[1]

Unit

An algebraic structure closed under two operations

Recovery means

Not applicable

Measured by

Checked by verifying that the ring axioms hold for the two operations

A closed loop of atoms bonded in sequence, so that the chain of bonds returns to its starting atom and forms a cycle within a molecule. The referent is a physical arrangement of atoms; benzene, the six-carbon aromatic ring, is the classic case, but rings run from strained three-membered cycles to large macrocycles, and their stability, shape, and reactivity follow from the bonding. This meaning shares nothing with the algebraic one but the word. Here the loop is literal, made of covalent bonds you could in principle count around the cycle, and the object is a single molecule rather than a set with operations.[2]

Unit

A cyclic arrangement of covalently bonded atoms

Recovery means

Not applicable

Measured by

Identified from molecular structure, by spectroscopy and X-ray crystallography

A flat band of countless particles, ice and rock ranging from dust grains to boulders, orbiting a planet in its equatorial plane. The referent is an astronomical disc of debris. Saturn's rings are the brightest example, held in place and sculpted by the planet's gravity and by small shepherding moons, while Jupiter, Uranus, and Neptune carry fainter systems too. This meaning keeps only the shape suggested by the word, a broad circle seen from far away. The ring is not one loop of bonded atoms but an uncounted swarm of separate bodies, and it is a feature of a planetary system rather than a molecule, a criminal network, or an algebraic structure.[3]

Unit

A band of orbiting particles around a planet

Recovery means

Not applicable

Measured by

Observed by reflected light, stellar occultation, and spacecraft imaging