SUMMARY / RELATED TOPICS

In abstract algebra, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring X, denoted by End. Addition of endomorphisms arises in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a ring, with the zero map 0: x ↦ 0 as additive identity and the identity map 1: x ↦ x as multiplicative identity; the functions involved are restricted to what is defined as a homomorphism in the context, which depends upon the category of the object under consideration. The endomorphism ring encodes several internal properties of the object; as the resulting object is an algebra over some ring R, this may be called the endomorphism algebra. An abelian group is the same thing as a module over the ring of integers, the initial ring. In a similar fashion, if R is any commutative ring, the endomorphism monoids of its modules form algebras over R by the same axioms and derivation. In particular, if R is a field F, its modules M are vector spaces V and their endomorphism rings are algebras over the field F.

These observations are the starting point for enriched category theory, as the categories Ab, R-Mod and F-Vect have hom functors valued in the categories of Z-, R- and F-algebras, are thus enriched in themselves. Let be an abelian group and we consider the group homomorphisms from A into A. Addition of two such homomorphisms may be defined pointwise to produce another group homomorphism. Explicitly, given two such homomorphisms f and g, the sum of f and g is the homomorphism:= f + g. Under this operation End is an abelian group. With the additional operation of composition of homomorphisms, End is a ring with multiplicative identity; this composition is explicitly:= f. The multiplicative identity is the identity homomorphism on A. If the set A does not form an abelian group the above construction is not additive, as the sum of two homomorphisms need not be a homomorphism; this set of endomorphisms is a canonical example of a near-ring, not a ring. Endomorphism rings always have additive and multiplicative identities the zero map and identity map.

Endomorphism rings are associative, but non-commutative. If a module is simple its endomorphism ring is a division ring. A module is indecomposable if and only if its endomorphism ring does not contain any non-trivial idempotent elements. If the module is an injective module indecomposability is equivalent to the endomorphism ring being a local ring. For a semisimple module, the endomorphism ring is a von Neumann regular ring; the endomorphism ring of a nonzero right uniserial module has either one or two maximal right ideals. If the module is Artinian, projective or injective the endomorphism ring has a unique maximal ideal, so that it is a local ring; the endomorphism ring of an Artinian uniform module is a local ring. The endomorphism ring of a module with finite composition length is a semiprimary ring; the endomorphism ring of a continuous module or discrete module is a clean ring. If an R module is finitely generated and projective the endomorphism ring of the module and R share all Morita invariant properties.

A fundamental result of Morita theory is that all rings equivalent to R arise as endomorphism rings of progenerators. In the category of R modules the endomorphism ring of an R-module M will only use the R module homomorphisms, which are a proper subset of the abelian group homomorphisms; when M is a finitely generated projective module, the endomorphism ring is central to Morita equivalence of module categories. For any abelian group A, M n ≅ End ⁡, since any matrix in M n carries a natural homomorphism structure of A n as follows: = ( ∑ i = 1 n φ 1 i ⋮ ∑ i =

Icelandic is a North Germanic language spoken by about 314,000 people, the vast majority of whom live in Iceland where it is the national language. It is most related to Faroese and Western Norwegian; the language is more conservative than most other Western European languages. While most of them have reduced levels of inflection, Icelandic retains a four-case synthetic grammar and is distinguished by a wide assortment of irregular declensions. Since the written language has not changed much, Icelanders are able to read classic Old Norse literature created in the 10th through 13th centuries with relative ease. Icelandic is related to, but not mutually intelligible when spoken with, the Faroese language whereas the written forms of the two languages are similar, it is not mutually intelligible with the continental Scandinavian languages, is farther away from the most spoken Germanic languages English and German than those three are. Aside from the 300,000 Icelandic speakers in Iceland, it is spoken by about 8,000 people in Denmark, 5,000 people in the United States, more than 1,400 people in Canada, notably in the region known as New Iceland in Manitoba, settled by Icelanders beginning in the 1880s.

The state-funded Árni Magnússon Institute for Icelandic Studies serves as a centre for preserving the medieval Icelandic manuscripts and studying the language and its literature. The Icelandic Language Council, comprising representatives of universities, the arts, journalists and the Ministry of Culture and Education, advises the authorities on language policy. Since 1995, on 16 November each year, the birthday of 19th-century poet Jónas Hallgrímsson is celebrated as Icelandic Language Day; the oldest preserved texts in Icelandic were written around 1100 AD. Many of the texts are based on poetry and laws traditionally preserved orally; the most famous of the texts, which were written in Iceland from the 12th century onward, are the Icelandic Sagas. They comprise the eddaic poems; the language of the sagas is a western dialect of Old Norse. The Dano-Norwegian later Danish rule of Iceland from 1536 to 1918 had little effect on the evolution of Icelandic, which remained in daily use among the general population.

Though more archaic than the other living Germanic languages, Icelandic changed markedly in pronunciation from the 12th to the 16th century in vowels. The modern Icelandic alphabet has developed from a standard established in the 19th century by the Danish linguist Rasmus Rask, it is based on an orthography laid out in the early 12th century by a mysterious document referred to as The First Grammatical Treatise by an anonymous author, referred to as the First Grammarian. The Rasmus Rask standard was a re-creation of the old treatise, with some changes to fit concurrent Germanic conventions, such as the exclusive use of k rather than c. Various archaic features, as the letter ð, had not been used much in centuries. Rask's standard constituted a major change in practice. 20th-century changes include the use of é instead of je and the removal of z from the Icelandic alphabet in 1973. Apart from the addition of new vocabulary, written Icelandic has not changed since the 11th century, when the first texts were written on vellum.

Modern speakers can understand the original sagas and Eddas which were written about eight hundred years ago. The sagas are read with updated modern spelling and footnotes but otherwise intact. With some effort, many Icelanders can understand the original manuscripts. According to an act passed by parliament in 2011, Icelandic is "the national language of the Icelandic people and the official language in Iceland". Iceland is a member of the Nordic Council, a forum for co-operation between the Nordic countries, but the council uses only Danish and Swedish as its working languages. Under the Nordic Language Convention, since 1987 Icelandic citizens have had the right to use Icelandic when interacting with official bodies in other Nordic countries, without becoming liable for any interpretation or translation costs; the convention covers visits to job centres, the police and social security offices. It does not have much effect since it is not well known, because those Icelanders not proficient in the other Scandinavian languages have a sufficient grasp of English to communicate with institutions in that language.

The Nordic countries have committed to providing services in various languages to each other's citizens, but this does not amount to any absolute rights being granted, except as regards criminal and court matters. Icelandic has minor dialectal differences phonetically; the language has both monophthongs and diphthongs, consonants can be voiced or unvoiced. Voice plays a primary role in the differentiation of most consonants including the nasals but excluding the plosives; the plosives b, d, g are voiceless and differ from p, t and k only by their lack of aspiration. Preaspiration occurs before geminate t and k, it does not occur before geminate d or g. Pre-aspirated tt is analogous ety

Clarence L. Maxwell, known as Gunplay Maxwell, was a late 19th-century Old West gunfighter and businessman from Boston, Massachusetts. Born the son of a hotel manager, Maxwell was involved in fights in his youth, he received a good education, but in 1875 he was involved in a bar room brawl that resulted in his shooting and killing a friend of his. Maxwell fled to Texas, Montana, to avoid being arrested for the murder. While in Montana and working as a cowboy, Maxwell began selling his gunman skills during the cattle-sheep wars, he became involved in cattle rustling in Wyoming and Utah, resulting in his being arrested and sentenced to three years in the Wyoming State Prison in 1893. While in prison, he became associated with Butch Cassidy, the two were released within a week of one another, it was said that he attempted to join Cassidy's gang, but was rejected. As to whether this is true or rumor is not confirmed. In any event, he formed his own gang, but it did not last and by 1898 he was riding alone again.

That year Maxwell and another man robbed the Springville, Utah bank, taking \$3,000. More than one hundred posse members pursued them, killing Maxwell's partner, capturing Maxwell after a brief shootout, he was taken to the Provo, Utah jail, but never revealed the identity of his partner, despite the latter being dead. Most of the money was recovered, either in the possession of the two robbers, or hidden near where the posse caught up to them. Maxwell was convicted of robbery, he was sent to the Utah State Prison. Five years after he helped to stop a prison escape, his sentence was commuted. While working as a mine guard, Maxwell discovered ozokerite near Colton and filed a claim started the mining company "Utah Ozokerite Company"; the mine soon became the largest known ozokerite mine in the world, Maxwell opened it to the public. Despite the success of his business venture, Maxwell for unknown reasons moved on, although it is believed he retained ownership of the mine, he surfaced in Goldfield, where he worked for mining companies by spying on striking miners.

While there, he killed a man named Joseph Smith during a dispute, but was not prosecuted. In 1907 he was involved in another gunfight in Utah, with a man named L. C. Reigle. Both were wounded. Maxwell was arrested, but again was not prosecuted; that year, in San Francisco, Maxwell married wealthy widow Bessie Hume, with whom he moved to Ogden, Utah. However, by June 1908, Maxwell was again on the move, while in the company of William M. Walters he robbed a Wells Fargo in Rawhide, Nevada. Both were captured, released on bail, were never brought to trial. On August 23, 1909, Maxwell confronted Deputy Sheriff Edward Black Johnstone in Price, tasked to stop a possible robbery that Maxwell had been planning. Maxwell allegedly, held a grudge against Johnstone due to the deputy having identified Maxwell as being a "bad man" and an ex-convict to the sheriff of Goldfield, Nevada. Maxwell confronted Johnstone in the Price Saloon, informed him that he intended to kill him; the dispute moved them out into the street, where Maxwell drew his pistol and opened fire, but missed.

Johnstone returned fire, knocking Maxwell to the ground. Maxwell attempted to fire yet again. Maxwell then said "Don't shoot again Johnstone, you have killed me." Maxwell died shortly thereafter. When his body was prepared for burial, it was discovered that his arms were covered with track marks, opium was found in his pocket, giving rise to the suspicion that he had become a drug addict, he was, at the time, going under the name William H. Seaman, was buried in the Salt Lake City Cemetery, his grave is today unmarked, making its location not certain. Clarence L. Maxwell Historical Accounts

The Ernest Hemingway International Billfishing Tournament is an annual fishing tournament held in Cuba. The tournament was established by American author Ernest Hemingway in 1950. Held in May or June, it has been described as the "highlight of Cuba's fishing year" and attracts anglers from as many as 30 countries. Hemingway won the first three editions of the tournament. Mina Hemingway, Ernest Hemingway's granddaughter, has twice won the tournament. In 1960, Fidel Castro won the event; as of 2016, the four-day tournament was sanctioned by the International Game Fish Association with competitors limited to 36 kilogram test line. Since 1997 it has been a release tournament; the event is based out of the Hemingway International Yacht Club in Havana

Hartley is a census-designated place in Hartley County, United States. The population was 540 at the 2010 census. In 1832, John Charles Beales and Jose Manuel Royella were granted the section where Hartley is now located under colonization laws of Mexico and Texas, they represented Texas Land Company. In 1875, the Texas Legislature passed an act which allowed contractors to clear one mile of the river Sabine and Neches, the Pine Island Bayou in exchange for land grants; the contractors were required to survey the land, return field notes to the Land Commissioner and request him to number the sections. The contractors received a deed to the uneven numbered sections with the State retaining the even-numbered ones as school lands; the field notes of Beaty and Forwood were filed November 17, 1875. The only other deeds registered in the county prior to 1884 were those of the Fort Worth and Denver Railroad and Burleson, Gunter and Burleson and W. M. Lee. On July 1, 1888, Beaty and Forwood sold Section 23 to James A.

Hudson of Logan County and John A. Lutz of New York for \$10,000.00. They sold the right of way to the Fort Worth and Denver Railroad July 1, 1888. Many changes occurred in a short time. Handbills were distributed throughout the eastern and southern states by the owners who were anxious to turn Hartley into a booming metropolis. Section 23 was registered with the Land Commissioner of Texas. A tent city came into being and grew as people came with high expectations and determination to make for themselves a new life in a new land. Hartley is located at 35°53′4″N 102°23′54″W. According to the United States Census Bureau, the CDP has a total area of 7.0 square miles, all of it land. As of the census of 2000, there were 441 people, 149 households, 118 families residing in the CDP; the population density was 63.2 people per square mile. There were 157 housing units at an average density of 22.5/sq mi. The racial makeup of the CDP was 92.29% White, 0.45% African American, 0.91% Native American, 0.23% Asian, 4.31% from other races, 1.81% from two or more races.

Hispanic or Latino of any race were 11.11% of the population. There were 149 households out of which 45.0% had children under the age of 18 living with them, 69.8% were married couples living together, 3.4% had a female householder with no husband present, 20.8% were non-families. 17.4% of all households were made up of individuals and 10.1% had someone living alone, 65 years of age or older. The average household size was 2.96 and the average family size was 3.31. In the CDP, the population was spread out with 34.7% under the age of 18, 8.4% from 18 to 24, 29.9% from 25 to 44, 18.1% from 45 to 64, 8.8% who were 65 years of age or older. The median age was 30 years. For every 100 females, there were 104.2 males. For every 100 females age 18 and over, there were 97.3 males. The median income for a household in the CDP was \$38,500, the median income for a family was \$45,000. Males had a median income of \$25,469 versus \$18,750 for females; the per capita income for the CDP was \$16,150. About 10.2% of families and 10.0% of the population were below the poverty line, including 10.5% of those under age 18 and 11.1% of those age 65 or over.

Hartley is served by the Hartley Independent School District. According to the Köppen Climate Classification system, Hartley has a semi-arid climate, abbreviated "BSk" on climate maps

Slamet A. Sjukur was the founding father of contemporary Indonesian music, he worked in Paris under Olivier Messiaen and Henri Dutilleux. He was a lecturer at IKJ but because of his unconventional ideas, he had to leave, he has been living in Jakarta and Surabaya as a freelance composer and music critic. Developing the idea of minimax in music, his compositions are "notable for their minimal constellation of sounds and for their numerological basis which indicate the composer’s interest in a new ‘ecology of music’"; this idea views limitation not as obstructions but as a challenge to work with a simple material, maximally. His honors include the Bronze Medal from the Festival de Jeux d’Automne in Dijon, the Golden Record from the Académie Charles Cros in France and the Zoltán Kodály Commemorative Medal in Hungary. More Gatra named him a Pioneer of Alternative Music and he was made an Officier de l’Ordre des Arts et des Lettres and a life member of the Akademi Jakarta; some of his prominent students include Soe Tjen Marching.

STAGE: Sangkuriang, mixed chorus, 1958. ORCHESTRAL: Õm, 14 strings, 1995; this issue of Balungan published by the American Gamelan Institute, includes Syukur's piece Tetabuhan Sungut for a cappella voices, using motifs from different styles of gamelan