Rings are circular bands of precious metal, worn on the fingers or toes as ornaments or symbols of high status or authority. They also represent marriage or betrothal. In mythology, rings are often associated with supernatural powers and can be used as talismans to protect the wearer or to achieve a certain outcome.
The word “ring” is derived from the Greek word raki which means “circle.” Ancient Greeks, including the ancient Egyptians, wore rings as ornaments or for decoration. The Egyptians, primarily, used signet rings. Some rings were used to hold individual stones. These were placed into a bezel, which was a flat table or flat table with shoulders supporting the ring. This was done in Hellenistic times. During the time of Roman civilization, bezels were usually made of vitreous pastes.
Although the word “ring” is still used, the idea of a ring is not confined to circles, but can be any geometric shape. However, since the nineteenth century, the traditional distinctions between ring types had mostly dissolved. Today, rings are worn as ornaments and as symbols of fidelity or betrothal. It is common to see rings made of gold and silver. Other metals and stones are used for decorative purposes.
Rings can be classified into several subtypes. The simplest of them are polynomials. There are also rings of integers and complex numbers. While some ring types have circular cross sections, others have square or rectangular cross sections. Most rings are composed of three parts: a central element, a subring of that central element, and a multiplication symbol.
For example, there is a ring of integers called the p-adic integers. There is also a ring of p-adic integers, which is a ring composed of p-adic integers, with a central element called Z. An idempotent element is an element that e 2 = e.
Another ring type is called a ring of commutative multiplication group. This is a ring with a central element (usually a nilpotent matrix) and elements that commute with every element in X.
The simplest commutative rings are called fields. Fields are formed by a set of natural numbers N with standard addition and multiplication. If any element of the ring is not zero, then the corresponding element is nilpotent.
One of the key contributors to the conceptualization of rings was Hilbert. He first formalized the idea as a generalization of Dedekind domains. Later, Dedekind and Heinrich Weber branched out into the use of modules.
Commutative algebra has become an important branch of ring theory. It has been influenced by algebraic number theory and algebraic geometry. By the twentieth century, commutative rings were being studied as a generalization of polynomial rings.
Since commutative rings are generally assumed to have multiplicative identity, a ring is defined to satisfy a collection of axioms. For example, if a ring of integers is defined to have a multiplicative identity, then the ring is isomorphic to a ring of integers if and only if there is a isomorphism between a ring of integers and a ring of integers.