OpenLLM-France/wikimedia huggingface dataset:
# Reduced ring

In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x$^2$ = 0 implies x = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced.
The nilpotent elements of a commutative ring R form an ideal of R, called the nilradical of R; therefore a commutative ring is reduced if and only if its nilradical is zero. Moreover, a commutative ring is reduced if and only if the only element contained in all prime ideals is zero.
A quotient ring R/I is reduced if and only if I is a radical ideal.
Let ${\mathcal {N}}_{R}$ denote nilradical of a commutative ring $R$. There is a functor $R\mapsto R/{\mathcal {N}}_{R}$ of the category of commutative rings ${\text{Crng}}$ into the category of reduced rings ${\text{Red}}$ and it is left adjoint to the inclusion functor $I$ of ${\text{Red}}$ into ${\text{Crng}}$. The natural bijection ${\text{Hom}}_{\text{Red}}(R/{\mathcal {N}}_{R},S)\cong {\text{Hom}}_{\text{Crng}}(R,I(S))$ is induced from the universal property of quotient rings.
Let D be the set of all zero-divisors in a reduced ring R. Then D is the union of all minimal prime ideals.
Over a Noetherian ring R, we say a finitely generated module M has locally constant rank if ${\mathfrak {p}}\mapsto \operatorname {dim} _{k({\mathfrak {p}})}(M\otimes k({\mathfrak {p}}))$ is a locally constant (or equivalently continuous) function on Spec R. Then R is reduced if and only if every finitely generated module of locally constant rank is projective.

## Examples and non-examples

* Subrings, products, and localizations of reduced rings are again reduced rings.
* The ring of integers Z is a reduced ring. Every field and every polynomial ring over a field (in arbitrarily many variables) is a reduced ring.
* More generally, every integral domain is a reduced ring since a nilpotent element is a fortiori a zero-divisor. On the other hand, not every reduced ring is an integral domain; for example, the ring Z[x, y]/(xy) contains x + (xy) and y + (xy) as zero-divisors, but no non-zero nilpotent elements. As another example, the ring Z × Z contains (1, 0) and (0, 1) as zero-divisors, but contains no non-zero nilpotent elements.
* The ring Z/6Z is reduced, however Z/4Z is not reduced: the class 2 + 4Z is nilpotent. In general, Z/nZ is reduced if and only if n = 0 or n is square-free.
* If R is a commutative ring and N is its nilradical, then the quotient ring R/N is reduced.
* A commutative ring R of prime characteristic p is reduced if and only if its Frobenius endomorphism is injective (cf. Perfect field.)

## Generalizations

Reduced rings play an elementary role in algebraic geometry, where this concept is generalized to the notion of a reduced scheme.
# Lepuia'i

Lepuia'i is one of four small villages on Manono Island in Samoa in central South Pacific Ocean. The village population is 183.
The other villages on Manono island are Apai, Faleu and Salua.
All the settlements on Manono Island fall within the political district of Aiga-i-le-Tai.
Manono island is one of three isles in the Apolima Strait separating the country's two main islands Upolu and Savai'i. The other islands in the strait are Apolima and the tiny uninhabited islet of Nu'ulopa.
# Pukarani (Bolivia)

Pukarani (Aymara pukara fortress, -ni a suffix to indicate ownership, "the one with a fortress", Hispanicized spelling Pocarani) is mountain in the Andes of Bolivia. It lies in the Oruro Department, Sajama Province, in the north of the Turco Municipality. Pukarani is situated northeast of the volcanic complex of Asu Asuni and Milluni and east of Chunkarani.
# Poso Kota

Poso Kota (pronunciation) is a district in Poso Regency, Central Sulawesi, Indonesia. Along with the neighboring districts of North and South Poso Kota, it makes up the capital region of Poso Regency, Poso, covering 56.57 km$^2$ (21.84 square miles) with a population of 49,334 in mid 2022.
With a population of 25,288 inhabitants, and a population density of 1,864 per km$^2$ in 2024, Poso Kota is the most densely-populated and most populous district in Poso Regency. With an area of only 11.16 km$^2$ (4.31 square miles), Poso Kota is also the smallest district in the regency, with only 0.16% of the total area of Poso Regency.

## Administrative division

As of 2024, Poso Kota consists of seven administrative villages (kelurahan).
* Gebangrejo
* Gebangrejo Barat
* Gebangrejo Timur
* Kayamanya
* Kayamanya Sentral
* Moengko
* Moengko Baru
# Navarretia squarrosa

Navarretia squarrosa (skunkbush, skunkweed, or California stinkweed) is a spreading annual plant from North America which is noted for its skunk-like odour. It grows to between 10 and 60 cm in height and has tubular lilac pink to deep blue flowers up to 12 mm in diameter in dense terminal heads, encircled by spiny sepals and bracts. The leaves are pinnately lobed and spiny.

## Occurrence

The species is native to British Columbia, Washington, Oregon, and California, and is naturalised elsewhere, including Australia and New Zealand. It is commonly found at elevations of less than 800 metres on open, wet, gravelly flats and slopes.
Specific plant associations where N. squarrosa occurs include the specialized Monterey Cypress forests near Carmel, California.
# Ollé

Ollé (French pronunciation:) is a commune in the Eure-et-Loir department in northern France.
# RewardVille

RewardVille is Zynga’s defunct virtual in-game currency and rewards program. Launched in March 2011 and retired on 5 December 2012, the program allowed players to earn exclusive Zynga-specific points and coins that could be spent within Zynga games or used to unlock virtual goods for use in the games or to gift to other players. Players collected “zCoins” currency and “zPoints”, which could be used to upgrade a user's Zynga status. Players could earn as many as 80 zPoints per game or 300 zPoints total in one day, by playing Zynga games.
RewardVille was available in French, Italian, German, Spanish, Portuguese, and English.
In a partnership with Lady Gaga, virtual items such as a Lady Gaga Gyrosphere in CityVille, a Lipstick Gun in Mafia Wars and a Paws Up Statue for FrontierVille were available in RewardVille. It was the first time Zynga had partnered with an artist to offer exclusive items.
In a partnership with Pizza Hut, a $5 or more donation to benefit the World Hunger Relief campaign gave players a choice of a Food Wagon for use in FarmVille, a Nutrition Center for use in CityVille, a Farm for use in Empires & Allies, or a Merino Sheep for use in The Pioneer Trail.
# Mulberry Street

Mulberry Street may refer to:

## Places

* Mulberry Street (Baltimore)
* Mulberry Street (Manhattan)
* Mulberry Street (Springfield, Massachusetts)
* Mulberry Street, Philadelphia, renamed Arch Street in 1854
* Mulberry Street Bridge, Harrisburg, Pennsylvania

## Arts and entertainment

* Mulberry Street (EP), by Cold War Kids, 2005
* Mulberry Street (film), a 2006 horror film
* "Mulberry Street", a song by Twenty One Pilots from Scaled and Icy, 2021
# Reduced product

In model theory, a branch of mathematical logic, and in algebra, the reduced product is a construction that generalizes both direct product and ultraproduct.
Let {S$_i$ | i ∈ I} be a nonempty family of structures of the same signature σ indexed by a set I, and let U be a proper filter on I. The domain of the reduced product is the quotient of the Cartesian product
    $\prod _{i\in I}S_{i}$
by a certain equivalence relation ~: two elements (a$_i$) and (b$_i$) of the Cartesian product are equivalent if
    $\left\{i\in I:a_{i}=b_{i}\right\}\in U$
If U only contains I as an element, the equivalence relation is trivial, and the reduced product is just the direct product. If U is an ultrafilter, the reduced product is an ultraproduct.
Operations from σ are interpreted on the reduced product by applying the operation pointwise. Relations are interpreted by
    $R((a_{i}^{1})/{\sim },\dots ,(a_{i}^{n})/{\sim })\iff \{i\in I\mid R^{S_{i}}(a_{i}^{1},\dots ,a_{i}^{n})\}\in U.$
For example, if each structure is a vector space, then the reduced product is a vector space with addition defined as (a + b)$_i$ = a$_i$ + b$_i$ and multiplication by a scalar c as (ca)$_i$ = c a$_i$.
# Leptysma

Leptysma is a genus of spur-throat toothpick grasshoppers in the family Acrididae. There are about five described species in Leptysma.

## Species

These five species belong to the genus Leptysma:
* Leptysma argentina Bruner, 1906
* Leptysma filiformis (Serville, 1838)
* Leptysma intermedia Bruner, 1911
* Leptysma marginicollis (Serville, 1838) (cattail toothpick grasshopper)
* Leptysma tainan Rehn & Hebard, 1938
# Pukara culture

The Pucará culture was an archaeological culture which developed in Qullaw, along the north-western shore of Lake Titicaca. It was characterized by a hierarchy of smaller centers and villages scattered throughout the northern basin of the Titicaca. The name originates from the town of Pukara, one of the largest settlements in the region. The modern town of Pucará is located half a mile to the east of the archaeological site. The Pukara culture is unrelated to the stone fortresses, pukaras, built across the Andes during the Inca Empire. Its sphere of influence reached as far north as the Cuzco Valley and as far south as Tiahuanaco. The culture had two phases of development within the Formative Period: the Middle Formative (1400 to 550 BC), and Late Formative (550 BC to 400 AD).

## Agriculture

The Pukara engaged in agriculture, herding, fishing, and domesticating the alpaca. Complex knowledge of hydraulics, and constructing ridges and furrows was required to ensure productive agriculture in the high-altitude environment. Q'ochas were natural and modified depressions in the earth that functioned as sunken gardens. Located near the Pucará River, the Pukara flooded the land with water before harvesting. Over a period of 3 to 5 years, agriculturalists would rotate between growing potatoes, quinoa, barley, and wheat. These strategies helped the Pukara establish settlements along the western slope of the Andes in the inter-Andean valley of Cuzco and Moquegua.

## Ceramics and sculpture

They developed, especially in the second phase, a very particular vigorous sculpture and ceramic culture. Pukara ceramics are painted in various colors. The fineware was painted with black and yellow paints on red clay and named the Classic Pukara Style. Incision lines define the shapes of humans, mythical creatures, tools, or shamanistic figures. Pukara pottery and textiles are found widely in the middle Andean, and the coastal Pacific valleys, reaching into Peru and Chile. Within the ceramic timeline of the Indigenous peoples of the Americas, the Classic Pukara Style is estimated to have peaked around 200 BCE, through 200 CE.

## The Pukara and Tiwanaku

The rise of Tiwanaku may have contributed to the weakening of the Pukara around 200 AD. The Pukara settlements were occupied by people from Tiwanaku. It's probable that the leaders of this social formation had taken the site of Pukara as a sacred place, and perhaps assumed it as their paqarina or place of origin. They copied Pucará's architectural model and commissioned a vast number of workers and engineers to transfer it to their capital in the Tiwanaku Valley, in the southern basin. It is possible that at that time, the sculptures of Pukará had become objects of great religious and ritual value, their possession being a symbol of power and prestige.
# Ralph Hazlett Upson

Ralph Hazlett Upson (June 21, 1888 – August 13, 1968) was a pioneer in the aviation field, holding Airship Pilot's Certificate #7, Balloon Pilot's Certificate #48 and Pilot's License #10290. Upson designed the world's only all-metal stressed-skin airship and contributed to aerospace technology research. He won the 1913 International Balloon Race (with Ralph Preston). He was the second individual to win the Wright Brothers Medal in 1929.

## Biography

He was born in New York City on June 21, 1888, to Grace Hazlett, a physician, and William Ford Upson, a Wall Street attorney. His brother William Hazlett Upson was a noted author. Upson graduated from the Stevens Institute of Technology in 1910 and began work in the aeronautical office at the Goodyear Tire and Rubber Company in Akron, Ohio. In the 1920s, he worked in the area of manned balloon flight at Goodyear. During World War I and immediately thereafter (1917-1921), Upson worked on the development of the US Navy's B, C, D, E, and F type airships and the US Army's AC and TC types. In 1920, he left Goodyear to work on development of a metalclad airship, becoming president (and chief engineer) of the Aircraft Development Corporation. The US Navy awarded a contract for one metalclad airship in 1928, and the next year Upson's concept of a duraluminum-skinned airship, the Detroit ZMC-2, launched from Detroit's Gross Ile Airport on 20 August 1929. The ZMC-2 remained in service until 1941, making 752 flights.
In the 1930s, Upson did consulting work for the US Navy, the National Advisory Committee for Aeronautics, and taught aerodynamics at the University of Michigan in Ann Arbor. He also designed the Union Pacific Railroad's City of Salina, a streamlined train designed with aerodynamic considerations. During World War 2, Upson was head of the aeronautics department at the Heinz Company, a builder of plywood gliders. He joined the staff of New York University in 1944 and moved in 1946 to the University of Minnesota, teaching aeronautical engineering until 1956. From 1956 to 1964, he was a research specialist at Boeing. Among his projects there was the aborted X-20 Dyna-Soar orbital space glider.
Upson had a heart attack while climbing Glacier Peak and died on August 13, 1968, at the age of 80 at Burien General Hospital in Burien, Washington.
# Navarretia sinistra

Navarretia sinistra (formerly Gilia sinistra) is a species of flowering plant in the phlox family known by the common name Alva Day's pincushionplant.

## Description

Navarretia sinistra produces a branching, leafy stem coated in knobby glands. The leaves are sometimes deeply cut or lobed.
The inflorescence produces generally 2 or 3 flowers on very thin stalks. Each flower has a pouchlike calyx of sepals which are ribbed with reddish membranous tissue between. The tubular flower has a pink corolla and a red-streaked yellow throat. The protruding stamens are tipped with blue anthers.
The bloom period is June to August.

## Distribution and habitat

The plant is endemic to the western United States, within northern California, Nevada, and Oregon.
It is native to mountain chaparral, sagebrush scrub, yellow pine forest, red fir forest, and lodgepole forest habitats, often on volcanic or serpentine soils. It grows at 50–2,700 metres (160–8,860 ft) in elevation.
# Olympic Museum (disambiguation)

Olympic Museum may refer to one of the following:
Museums linked to the Olympic Games:
* Olympic Museum, in Lausanne, Switzerland
* Australian Gallery of Sport and Olympic Museum, Melbourne, Australia
* Juan Antonio Samaranch Olympic and Sports Museum, in Barcelona, Spain
* Lake Placid Winter Olympic Museum, in Lake Placid, New York
* Norwegian Olympic Museum, in Lillehammer, Norway
* Seoul Olympic Museum, in Seoul, South Korea
* Thessaloniki Olympic Museum, in Thessaloniki, Greece
* British Olympic Museum, is planning to be built in London, Great Britain
* New Zealand Olympic Museum, in Wellington New Zealand
* Sarajevo Winter Olympics Museum, in Sarajevo, Bosnia and Herzegovina
* Olympic Museum in Cotonou, Benin, belonging to the Comité olympique et sportif béninois, at the Stade de l'Amitié-Kouhounou
* United States Olympic & Paralympic Museum, in Colorado Springs, Colorado
Museums linked to Olympia, Washington, USA:
# Wubana atypica

Wubana atypica is a species of sheetweb spider in the family Linyphiidae. It is found in the United States and Canada, and was first described in 1936 by Chamberlin and Irvie.
# Museu de Arte de Ribeirão Preto

The Museu de Arte de Ribeirão Preto "Pedro Manuel-Gismondi" (Art Museum of Ribeirão Preto or Museum Pedro Manuel Gismondi, short MARP) is an art museum in the city of Ribeirão Preto in the state of São Paulo, Brazil. Opened in 1992, with the aim of bringing together the art collection belonging to the city, the museum is a local public institution, subordinated to the Municipal Secretary of Culture. Since its founding, the museum is housed in the former Recreation Society building, opened in 1908, who has also served as headquarters to the City of Ribeirão Preto.
In less than two decades of existence, the museum has hosted major exhibitions and competitions and has developed projects of national scope, highlighting the Art Salon of Ribeirão Preto (SARP). The museum has an art collection focusing on regional production, but also including important names in modern and contemporary art in Brazil.
It is equipped with a specialized library and maintains extensive educational activities and cultural. The museum also runs one second exhibition space, convention center located in Ribeirão Preto, and develops its activities in close collaboration with other cultural institutions of the city.
# Red neusticurus

There are two species of lizard named red neusticurus:
* Neusticurus rudis, native to Guyana and Venezuela
* Neusticurus surinamensis, native to Brazil
# Ministry of Local Government and Rural Development

Ministry of Local Government and Rural Development may refer to:
* Ministry of Local Government, Rural Development and Co-operatives (Bangladesh)
* Ministry of Local Government and Rural Development (Botswana)
* Ministry of Local Government and Rural Development (Ghana)
* Ministry of Local Government and Rural Development (Pakistan)
* Ministry of Local Government and Rural Development (Sierra Leone)
* Ministry of Local Government and Rural Development (Zambia)
# Texila American University Zambia

Texila American University Zambia is a private university in Lusaka, Zambia. It received its academic charter from the Higher Education Authority of Zambia in 2016 and is part of the Texila American University Consortium. The university offers several bachelor's degrees, master's degrees, and PhD programs, principally in the medical sciences.

## Programs offered and accreditation

TAU Zambia offers undergraduate and postgraduate programs in fields such as medicine, public health, business administration, and information technology. The university emphasizes a curriculum that combines theoretical knowledge with practical experience. The university is recognized by relevant educational authorities in Zambia, ensuring that its programs meet national and international academic standards.

## Collaboration

TAU Zambia collaborates with local hospitals and healthcare institutions to provide students with hands-on clinical training, essential for their professional development in medical and health sciences.

## Texila American University Consortium (TAU – C)

Texila American University Consortium is a private educational group of universities and campuses. Incorporated in 2010, the TAU Consortium has campuses in Guyana, Zambia and the United Arab Emirates, with additional offices in India, the United States, Hong Kong and the Philippines. Its associated ventures include:
* Texila American University
* Texila American University Fujairah
* Texila e-Journal
* Texila e-Conference
* Texila e-Learning
# Taberi River

The Taberi River is a river on the Caribbean island of Dominica.
# Paul Thomas Sr. Field

Paul Thomas Field is a 1,200-seat baseball stadium in Wenatchee, Washington. It is located on the campus of Wenatchee Valley College; the college's baseball team shares the stadium with the Wenatchee AppleSox of the West Coast League.
540 of the stadium's seats were previously used at the Kingdome in Seattle.
# Rain (2008 film)

Rain is a 2008 Bahamian drama film directed by Maria Govan. This was among the first feature productions to be shot entirely by citizens of the Bahamas. Rain is among the final films of actor Calvin Lockhart.

## Plot

A young Bahamian girl, Rain, boards a local mail boat from Ragged Island and sets sail for Nassau. The death of her grandmother has forced her to get out and explore the world on her own. When she arrives in Nassau, the sights of the big city overwhelm her, and soon she finds her idealistic illusions shattered when she finds how destructive her mother's lifestyle has truly become. Stranded in an unfamiliar environment that fills her with dread and confronted by a mother she has never known, Rain searches desperately to find her own place in the world.

## Production

Rain was the first film to be directed by Maria Govan, who raised the US$1 million budget through Bahamian investors. It was also one of the first features to be produced entirely in the Bahamas (the first being "Filthy Rich Gangster"), which had long been a filming location for the James Bond films among other efforts. Govan's film was shot during four weeks on three different islands of the Bahamas, including New Providence and Eleuthera; filming also took place in New York City.

## Release

In early September 2008, Rain received its world premiere at the Toronto International Film Festival. On December 4, 2008, it became the opening selection of the 2008 Bahamas International Film Festival.
Rain was released on DVD in the U.S. on July 27, 2010.

## Reception

When the film was shown at the Toronto Film Festival, Alissa Simon of Variety said: "Displaying an eye for composition, helmer Govan (who spent two years observing a crack house for docu Where I’m From: HIV and AIDS in the Bahamas), knows her milieu — essentially the antithesis of Bahamian tourist-board images. Martina Radwan’s gritty lensing and Denise Hudson’s detailed production design provide ample support." She, however, commented on "the script's melodramatic cliches and some weak supporting [performances]".
# She's Been Away

"She's Been Away" is a 1989 British television film written by Stephen Poliakoff and directed by Sir Peter Hall. In her final appearance it starred Dame Peggy Ashcroft, who won two awards at the Venice International Film Festival for her performance, as did Geraldine James. It was nominated for the BAFTA TV Award for Best Single Drama.

## Plot

The closure of a mental institution threatens to leave the elderly Lillian homeless. Her wealthy nephew Hugh takes her in, putting additional strain on his wife Harriet. Gradually, an awkward friendship develops between Harriet, on the verge of a nervous breakdown herself, and Lillian, who has spent fifty years as a mental patient.

## Cast

* Peggy Ashcroft as Lillian Huckle
* Geraldine James as Harriet Ambrose
* James Fox as Hugh Ambrose
* Jackson Kyle as Dominic Ambrose
* Rebecca Pidgeon as Young Lillian
* Rosalie Crutchley as Gladys
* Rachel Kempson as Matilda
* Cryss Jean Healey as Young Margaret
* Leslie Goodall as Old Edward
* Edgar Goodall as Old Thomas
* René Zagger as Young Thomas
* Barnaby Holm as Young Edward
* Donald Douglas as Lillian's father
* Hugh Lloyd as George
* Brid Brennan as Lillian's nurse
* David Hargreaves as Lillian's doctor
* Hugh Ross as 1920s doctor
* Carrie Thomas as 1920s nurse
# Rex MacLeod

John Alexander "Rex" MacLeod (March 30, 1921 – April 7, 1995), was a Canadian sports journalist. A columnist for The Globe and Mail and Toronto Star, he won the Elmer Ferguson Memorial Award in 1987 and is a member of the media section of the Hockey Hall of Fame. MacLeod served in World War II with the Royal Canadian Air Force and entered the newspaper industry after the war, with the Guelph Mercury. He retired in 1987.