What is ring theory in mathematics?
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Commutative rings are much better understood than noncommutative ones.
What is meant by ring in discrete mathematics?
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers.
What is the difference between an algebra and a ring?
As a vague summary, the algebraic structure of a ring is entirely internal, but in an algebra there is also structure coming from interaction with an external ring of scalars.
What is ring explain properties of ring?
A ring is a set equipped with two operations (usually referred to as addition and multiplication) that satisfy certain properties: there are additive and multiplicative identities and additive inverses, addition is commutative, and the operations are associative and distributive.
Why is it called the ring?
Kôji Suzuki, the author of the novel upon which the movies are based, says that the title referred to the cyclical nature of the curse, since, for the viewer to survive after watching it, the video tape must be copied and passed around over and over.
Is every algebra a ring?
Therefore a ring can be regarded as a special case of an algebra. If A is an algebra over a field Φ, then, by definition, A is a vector space over Φ and therefore has a basis.
Why are rings called rings in math?
The name “ring” is derived from Hilbert’s term “Zahlring” (number ring), introduced in his Zahlbericht for certain rings of algebraic integers. As for why Hilbert chose the name “ring”, I recall reading speculations that it may have to do with cyclical (ring-shaped) behavior of powers of algebraic integers.
What is ring in linear algebra?
ring, in mathematics, a set having an addition that must be commutative (a + b = b + a for any a, b) and associative [a + (b + c) = (a + b) + c for any a, b, c], and a multiplication that must be associative [a(bc) = (ab)c for any a, b, c].
What is the purpose of rings?
Although some wear rings as mere ornaments or as conspicuous displays of wealth, rings have symbolic functions respecting marriage, exceptional achievement, high status or authority, membership in an organization, and the like.
What is ring theory used for?
Ring Theory is an extension of Group Theory, vibrant, wide areas of current research in mathematics, computer science and mathematical/theoretical physics. They have many applications to the study of geometric objects, to topology and in many cases their links to other branches of algebra are quite well understood.
What does “let be a local ring” mean?
We often say “let be a local ring” to indicate that is local, is its unique maximal ideal and is its residue field. A local homomorphism of local rings is a ring map such that and are local rings and such that .
What is the difference between local ring and unique maximal ideal?
The unique maximal ideal consists of all multiples of p. More generally, a nonzero ring in which every element is either a unit or nilpotent is a local ring. An important class of local rings are discrete valuation rings, which are local principal ideal domains that are not fields.
What is the quotient ring of a local ring?
If $ A $ is a local ring with maximal ideal $ \\mathfrak m $, then the quotient ring $ A / \\mathfrak m $ is a field, called the residue field of $ A $. Examples of local rings. Any field or valuation ring is local.
What are the equivalent properties of a local ring?
A ring R is a local ring if it has any one of the following equivalent properties: R has a unique maximal left ideal. R has a unique maximal right ideal. 1 ≠ 0 and the sum of any two non- units in R is a non-unit. 1 ≠ 0 and if x is any element of R, then x or 1 − x is a unit.