Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series and functions. A ring is an abelian group with a second binary operation, associative, is distributive over the abelian group operation, has an identity element. By extension from the integers, the abelian group operation is called addition and the second binary operation is called multiplication. Whether a ring is commutative or not has profound implications on its behavior as an abstract object; as a result, commutative ring theory known as commutative algebra, is a key topic in ring theory. Its development has been influenced by problems and ideas occurring in algebraic number theory and algebraic geometry. Examples of commutative rings include the set of integers equipped with the addition and multiplication operations, the set of polynomials equipped with their addition and multiplication, the coordinate ring of an affine algebraic variety, the ring of integers of a number field.

Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators in the theory of differential operators, the cohomology ring of a topological space in topology. The conceptualization of rings was completed in the 1920s. Key contributors include Dedekind, Hilbert and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they proved to be useful in other branches of mathematics such as geometry and mathematical analysis; the most familiar example of a ring is the set of all integers, Z, consisting of the numbers …, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, …The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings. A ring is a set R equipped with two binary operations + and · satisfying the following three sets of axioms, called the ring axioms R is an abelian group under addition, meaning that: + c = a + for all a, b, c in R. a + b = b + a for all a, b in R.

There is an element 0 in R such that a + 0 = a for all a in R. For each a in R there exists −a in R such that a + = 0. R is a monoid under multiplication, meaning that: · c = a · for all a, b, c in R. There is an element 1 in R such that a · 1 = a and 1 · a = a for all a in R. Multiplication is distributive with respect to addition, meaning that: a ⋅ = + for all a, b, c in R. · a = + for all a, b, c in R. As explained in § History below, many authors follow an alternative convention in which a ring is not defined to have a multiplicative identity; this article adopts the convention that, unless otherwise stated, a ring is assumed to have such an identity. Authors who follow this convention sometimes refer to a structure satisfying all the axioms except the requirement that there exists a multiplicative identity element as a rng and sometimes as a pseudo-ring. For example, the set of integers with the usual + and ⋅ is a rng, but not a ring; the operations + and ⋅ are called multiplication, respectively.

The multiplication symbol ⋅ is omitted. Although ring addition is commutative, ring multiplication is not required to be commutative: ab need not equal ba. Rings that satisfy commutativity for multiplication are called commutative rings. Books on commutative algebra or algebraic geometry adopt the convention that ring means commutative ring, to simplify terminology. In a ring, multiplicative inverses are not required to exist. A nonzero commutative ring in which every nonzero element has a multiplicative inverse is called a field; the additive group of a ring is the ring equipped just with the structure of addition. Although the definition assumes that the additive group is abelian, this can be inferred from the other ring axioms; the proof makes use of the "1". Some authors define a ring without the requirement of associativity for multiplication; this general definition of a ring is useful in the sense that every algebra is a ring. Some basic properties of a ring follow from the axioms: The additive identity, the additive inverse of each element, the multiplicative identity are unique.

For any element x in a ring R, one has x0 = 0 = 0x and x = –x. If 0 = 1 in a ring R R has only one element, is called the zero ring; the binomial formula holds for any commuting pair of elemen

Jermaine Crumpton

Jermaine Crumpton is an American basketball player for T71 Dudelange of the Total League in Luxembourg. He played college basketball for the Canisius College and was named the 2018 Metro Atlantic Athletic Conference Co-Player of the Year; the Niagara Falls, New York native small forward played basketball at Niagara Falls High School and committed to Canisius in Buffalo as a junior. After sitting out the 2013–14 season as a redshirt, Crumpton was an immediate contributor, averaging 7.2 points in 17.6 minutes per game. As a junior, he averaged 15.9 points and 4.5 rebounds per game and was a third-team All-Metro Atlantic Athletic Conference selection. In the off-season prior to his senior year, Crumpton underwent a weight training regimen and lost 30 pounds, resulting in an ability to be more active and improve endurance; the plan worked well, as at the close of the 2017–18 season, was named the MAAC co-Player of the Year with Niagara's Kahlil Dukes. The duo were named honorable mention All-Americans by the Associated Press.

Crumpton signed his first professional deal with T71 Dudelange in Luxembourg’s Total League for the 2018–19 season. Canisius Golden Griffins bio College stats @


Sennyo-ji is a Shingon temple in Itoshima, Fukuoka Prefecture, Japan. Its honorary sangō prefix is Sennyo-ji Daihiō-in, it is referred to as Raizan Kannon. According to the legend, Sennyo-ji was founded in the Nara period by Seiga, who came from India as a priest during the period. Due to its position in the north overlooking the Sea of Genkai, it has been expected from the shogunate as a prayer temple of the foremost line against the Mongol invasions of Japan during the Kamakura period. In its heyday has been said to be lined up to 300 priest living quarters around the temple. Sennyo-ji is a general term of this temple, it is referred to as the priest's lodge, located next to the middle sanctuary, the present day site of Ikazuchi-jinja; the wooden Avalokiteśvara statue is the subject of mountainous faith, enshrined in the main hall. Afterwards the priest living quarters were ruined during the long war between Muromachi and Sengoku periods, there only remains the priest's lodge. In 1573, the main hall was founded by Kuroda Tsugutaka, the 6th feudal lord of Kuroda clan.

Big maple trees, designated as a natural monument of Fukuoka prefecture, has been said to be planted by him. Mount Rai has one at the middle of the mountain and one at its peak; the middle sanctuary was founded in honor of Emperor Suinin whom he is conventionally considered to have reigned from 29 BC to AD 70. Both sanctuaries have been governed by the temple until the Edo period. However, the priest's lodge in the middle sanctuary was abolished by the separation of Shinto from Buddhism, introduced after the Meiji Restoration. Cultural properties, such as Buddha statues including the main Buddha and the ancient documents, were moved to the main hall; the temple is known for being a great place for cherry blossom viewing in the spring, many people visit in the autumn to see the fall foliage. The 4.8 meter wooden Avalokiteśvara statue in the main hall is the work of the Kamakura period. Mount Rai Ikazuchi-jinja Official site