The Conceptualization of Rings in Algebra


Rings are a set of algebraic structures. They can be circular, semicircular, square or rectangular in cross section. The simplest ring is an integer ring. However, a commutative ring can be considered a nonzero ring with a multiplicative inverse. These inverses can exist, but are not required to make a ring commutative.

Rings are usually made from precious metals. Ancient Greeks used rings for decoration. Later, Rome used vitrous pastes. In the 3rd century bc, wearing gold rings was restricted to a few classes of people. During the Hellenistic period, bezels were introduced, which held individual stones. Modern rings may have geometric designs etched on them.

Several important contributions to the conceptualization of rings include Fraenkel, Hilbert, Noether and Dedekind. In commutative algebra, the main branch of ring theory, ring is seen as an algebraic structure. A ring is said to be commutative when its two binary operations are multiplication and addition. The simplest commutative ring is called a field.

The commutative ring of rational, real and complex numbers is known as the ring of integers. A commutative ring of quadratic integers is a subring of the ring of all algebraic integers. The ring of integers has two operations, namely, standard addition and multiplication.

A ring of all algebraic integers is also referred to as a ring of p-adic integers. The completion of Z at the principal ideal (p) is called Zp. It is generated by the prime number p. The ring of p-adic integers can be constructed from the p-adic absolute value on Q.

Rings are also generalizations of Dedekind domains. They are defined to have an axiomatic definition. The term “ring” is also applied to other structures, including rings of differential operators. A ring of n x n real square matrices with n>=2 is known as a group ring. These rings can be thought of as preadditive categories with a single object, and have their own homomorphisms.

A commutative ring is a ring whose elements form a commutative multiplication group. It is often viewed as a type of coordinate ring in affine algebraic variety. The simplest commutative matrices are polynomials and square real matrices.

The ring is a symbol of authority, marriage and fidelity. It can be worn on the finger or a toe. It is believed that the symbolism behind a ring can tell about the person’s character. Some people wear rings as ornaments, while others wear them to indicate their profession or relationship status. The ring has been associated with Rudyard Kipling. The sequels of The Ring were released thirteen years ago.

In commutative algebra, the ring has been influenced by the study of algebraic number theory and algebraic geometry. The ring of integers is the simplest commutative ring and contains the multiplicative identity element 1. This is one of the first rings to be formalized. It was the subject of numerous pathological examples, which motivated the examination of the Noetherian ring and its definition as an excellent ring.