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In mathematics commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian. If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is "not too large", in the sense that if R is Noetherian, the same must be true for R. Formally, Hilbert's Basis Theorem. If R is a Noetherian ring R is a Noetherian ring. Corollary. If R is a Noetherian ring R is a Noetherian ring; this can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the set of common roots of finitely many polynomial equations. Hilbert proved the theorem in the course of his proof of finite generation of rings of invariants. Hilbert produced an innovative proof by contradiction using mathematical induction. One can determine basis polynomials using the method of Gröbner bases. Theorem. If R is a left Noetherian ring the polynomial ring R is a left Noetherian ring. Remark. We will give two proofs, in both only the "left" case.

Suppose a ⊆ R is a non-finitely generated left-ideal. By recursion there is a sequence of polynomials such that if b n is the left ideal generated by f 0, …, f n − 1 f n ∈ a ∖ b n is of minimal degree, it is clear, a non-decreasing sequence of naturals. Let a n be the leading coefficient of f n and let b be the left ideal in R generated by a 0, a 1, …. Since R is Noetherian the chain of ideals ⊂ ⊂ ⊂ ⋯ must terminate, thus b = for some integer N. So in particular, a N = ∑ i < N u i a i, u i ∈ R. Now consider g = ∑ i < N u i X deg ⁡ − deg ⁡ f i, whose leading term is equal to that of f N. Let a ⊆ R be a left-ideal. Let b be the set of leadin

Bob Stanley is a British musician, journalist and film producer. He is a member of the indie pop group Saint Etienne and has had a parallel career as a music journalist, writing for NME, Melody Maker, The Guardian and The Times, as well as writing two books on music and football, he has a career as a DJ and as a producer of record labels, has collaborated on a series of films about London. His second publication, Yeah! Yeah! Yeah!: The Story of Modern Pop, was published by Faber & Faber in 2013. Stanley is a member of the group Saint Etienne for which he co-writes songs and produces. Live on stage, he plays keyboards. Stanley was educated at Whitgift School in London. After leaving school, Stanley worked in various record shops. While working at Virgin Records in Peterborough he met Andrew Midgley; the two produced a fanzine called Pop Avalanche in 1986. Stanley wrote four issues of Caff, a fanzine created with childhood friend Pete Wiggs. In 1987, Stanley sent an issue of Caff to James Brown live reviews editor for NME.

This led to Stanley's first commissioned work, a review of a Johnny Cash show in Peterborough. After two years he moved to Melody Maker, where he wrote until Saint Etienne became a full-time occupation in 1991; as Saint Etienne dominated his career, Stanley continued to write for The Face and Mojo in the 1990s. In the 2000s he has returned to journalism, writing about architecture as well as music, he contributes to various publications including The Times and The Guardian. In 2007, with Paul Kelly Stanley edited a book of football programme artwork. In 2013, Faber and Faber published Stanley's second book,Yeah Yeah Yeah: The Story of Modern Pop, a history of pop music from the publication of the first British pop chart in 1952 until the advent of iTunes in 2001. An abridged edition was published by W W Norton in the USA in 2014. In 2015, Turner translated into Spanish,Yeah Yeah Yeah: La historia del pop moderno, Spain, ISBN 978-84-16142-22-4 Stanley is working on his third book, Too Darn Hot: The Story of Popular Music.

This is scheduled to be published by Faber and Faber in 2020. This book looks at the history of popular music from the start of recorded music until the advent of rock n'roll in the early 1950s, he was the winner of the 2017 Eccles British Library Writers in Residence Award which supports his research for Too Darn Hot using the Library's American collections. While recording the album Finisterre in 2002, Pete Wiggs and frequent collaborator Paul Kelly made a film to accompany the record titled Finisterre, described by The Observer as a "cinematic hymn to London", it premiered at the Institute of Contemporary Arts in London and was screened around the world by one dot zero. In 2005 Saint Etienne and Kelly were invited by the Barbican Centre to create a film and music event, for which they made What Have You Done Today Mervyn Day, a drama-documentary set in the Lower Lea Valley, the site for the 2012 Olympic Games. In 2007 their third London film, This Is Tomorrow, a history of the Southbank Centre, premiered with a live performance, including a 60-piece orchestra, at the Royal Festival Hall.

Kelly and Saint Etienne collaborated again on How We Used to Live, described as "a cherishable, woozy-hazy trawl of London from postwar days to yuppiedom". Stanley has curated several film seasons for arts institutions including the Barbican, including Gonna Make You A Star and Britain Learns to Rock. In 2016, he was commissioned by 14-18 Now as creative producer on a project to explore the impact of the First World War on the north-east of England; the resulting film and music commission, featuring a film directed by artist-filmmaker Esther Johnson, co-produced and scripted by Stanley, a soundtrack by Field Music and Warm Digits, premiered at the Sunderland Empire in July 2016 and toured to the Barbican in London. In 2017, as part of Hull 2017: City of Culture's Mind on the Run season exploring the influence and legacy of jazz composer Basil Kirchin, Stanley co-directed a short film, Abstractions of Holderness, filmed in the isolated area of the east coast of England where Kirchin settled in the 1970s.

Pete Wiggs composed the soundtrack, performed at the Mind on the Run concert by the BBC Concert Orchestra and various musicians who had collaborated with Kirchin in the past. In the late 1980s/early 1990s, Stanley ran a record label called Caff, which released 17 7" singles, all limited to 500 copies, including early singles by the Manic Street Preachers and Pulp. Between 1992 and 1994, Stanley and Saint Etienne bandmate Pete Wiggs ran the indie label Icerink Records. In 1996, Stanley ran EMIDisc, again alongside Wiggs, backed by EMI Director of A&R Tris Penna; the label was to be an EMI sub-label devoted to new talent. The label was short-lived, releasing albums by Denim. Stanley and Wiggs previously ran a CD imprint called Eclipse through Universal. Stanley started his own imprint, Croydon Municipal, via Cherry Red in 2012, specialising in music from the mid-twentieth century. Stanley is known for his large collection of vinyl records; when Saint Etienne are between projects, he DJs, playing 1960s and 1970s pop music and soul.

With Wiggs, he ran. Cherrybomb, a girl group night in Bloomsbury, ran from 2006–2009, he works as a consultant for reissue record labels, notably Ace Records. English Weather, which he compiled

The Granite Chief Wilderness is a 19,048 acre federally designated wilderness area of the Tahoe National Forest. Created by the California Wilderness Act of 1984, it is located in the Sierra Nevada mountains west of Lake Tahoe in California, USA, it is managed by the US Forest Service Tahoe National Forest. Elevations range from 4,800 feet to 9,019 feet at the summit of Granite Chief. Events such as the Western States Endurance Run and the equestrian Western States Trail Ride, cross portions of the wilderness; the Pacific Crest Trail passes through along the east edge of the wilderness. This region is extensively glaciated and has features such as hanging valleys, cirques and U-shaped valleys, but few lakes. Just outside the wilderness boundary there are two large recreation reservoirs, Hell Hole Reservoir to the south and French Meadows Reservoir to the west; the Sierra Club had maintained the Bradley Hut, a ski hut located at the Five Lakes Basin, but in 1994 the Sierra Club was asked to remove the hut by the U. S. Forest Service as it was now inside the newly created wilderness.

The hut was relocated four miles away. Because no mechanical equipment can be used in a wilderness, the dismantling of the Bradley Hut took until the fall of 1996 to finish; the principal drainages are the Middle Fork of Five Lakes Creek. The small lakes within the wilderness boundary are the Five Lakes, Mildred Lake and Little Needle Lake. Fish such as rainbow and brown trout can be seen in Whiskey and Bear Pen creeks as well as the largest lake of the Five Lakes group. Rich, volcanic soils support a range of plant life, from fields of mule ears to conifer forests including whitebark pine at the highest elevations. Along the creeks grow black cottonwood and aspen; the three bracted onion is a native perennial bulb endemic to California. The California Native Plant Society lists the three bracted onion as " rare, threatened, or endangered... " and there are only 10 counties with either specimens obtained or a verified observation made. Near the North Forth American River is Whitney's milk vetch.

Typical of the high Sierra Nevada Mountains, the wildlife include mountain lion, black bear, mule deer. Granite Chief wilderness provides important fawning areas for mule deer, so visitors are prohibited from bringing dogs into certain areas of the wilderness from May 15 to July 15. Activities include day-hiking, fishing, cross-country skiing, mountain climbing and horsepacking; the Five Lakes basin is the most used area in the wilderness due to the close proximity to both Squaw Valley and Alpine Meadows ski areas. Two commercial outfitters offer horsepacking trips into the wilderness. There are 37 miles of trails with eight trailheads; the most used is the Granite Chief trailhead located at Squaw Valley Ski Area parking lot. Talbot campground is located near the Talbot trailhead, four miles north of French Meadows Reservoir and is the only trailhead with a no-fee campground. Tahoe National Forest Basques in the United States Text of the California Wilderness Act of 1984 Tahoe National Forest - Granite Chief Wilderness Wilderness.net TopoQuest map "Granite Chief".

SummitPost.org. Retrieved 2011-08-19

Shah Ghazi Rustam, was king of the Bavand dynasty of Mazandaran, ruling from 1142 to 1165. He expanded the borders of the kingdom at the expense of his neighbors the Ismailis and the Seljuks, he established a Bavandid presence in Gilan as a result of his frequent vengeful raids against the Ismailis, who had assassinated his son and heir, Girdbazu. He brought Qumis and Ray under Bavandid control during his wars against the Seljuks and the Karakhanids. Shah Ghazi's reign represented the pinnacle of Bavandid power and influence in Iran, Shah Ghazi himself was considered the most illustrious king of the dynasty; the name of Shah Ghazi Rustam is combination of Persian and Arabic—"shah" meaning king in Persian, "ghazi" meaning warrior in Arabic. "Rustam" was the name of the popular mythological Iranian warrior Rostam. Shah Ghazi Rustam's laqab was Nusrat al-Din. Shah Ghazi was born. 1105, as the son of Ali I, whose father, Shahriyar IV, was the Bavandid king of Mazandaran. The Bavand kingdom was during this period a vassal of the Seljuk Empire, which had a decade earlier under sultan Malik-Shah I controlled a vast area stretching from the Hindu Kush to eastern Anatolia and from Central Asia to the Persian Gulf.

However, after the assassination of Malik-Shah and his vizier Nizam al-Mulk in 1092, the Seljuk Empire had fallen into decline. Shahriyar IV had thus been able to disobey the orders of the Seljuk sultan Muhammad I Tapar several times. After Shahriyar IV's death in 1114, his son Qarin III succeeded him, started arresting and imprisoning many loyal servants of his father, thus weakening the kingdom, he fell ill, died in 1117 after he asked the local people to pledge allegiance to his son Rustam III, who succeeded him. Rustam III's reign, was more shortlived—he was poisoned by his stepmother, the sister of Muhammad I Tapar, who wanted to marry Ali I, who ascended the Bavandid throne. Shah Ghazi is first mentioned in 1119, when the Seljuk sultan Ahmad Sanjar ordered Ali to meet him at his court, who disobeyed, instead sent Shah Ghazi. Sanjar, was unsatisfied with Shah Ghazi and dismissed him after four months. In 1131, Sanjar ordered Ali to join him in a expedition to Iraq. Shah Ghazi was once again sent to Sanjar, where he distinguished himself and was wounded at the battle of Dinavar in 25 May 1132, where Sanjar emerged victorious.

Sanjar granted Shah Ghazi several honors and allowed him to return to his father in Mazandaran. In 1141/1142, the Khwarazm-Shah Atsiz seized Gorgan from Ali, he imprisoned the local Bavandid governor Rustam Kabudjama. Shah Ghazi, without the agreement of his father met Atsiz, persuaded him to release Rustam Kabudjama. Ali, after hearing about his son meeting with Atsiz without his approval, criticized his actions; some time Ali, too old to rule, abdicated in favor of Shah Ghazi. Ali died three years at Tammisha and was buried in Sari. Shah Ghazi's coronation was most done in traditional Bavandid fashion. In accordance with the ancient Iranian style, the coronation lasted seven days, included the typical banquets, giving of gifts, whilst the renowned statesmen, local rulers, members of the royal house gathered from all the landscapes. On the eighth day, after the felicitations were complete, Shah Ghazi ascended the throne, fastened the royal waistband, confirmed the governors in their offices, his antagonistic brother Taj al-Muluk Mardavij had been serving Sanjar at Marv, as a result had become appreciated by him receiving his sister in marriage.

After Ali's death, Sanjar gave Mardavij an army to help him become the ruler of the Bavandid kingdom. Mardavij proceeded to capture Gorgan and the fort of Johayna, not long after besieged Qal'a-ye Dara, where Shah Ghazi had fortified himself; the siege proved to be unsuccessful. Shah Ghazi died on 23 January 1165 after having been indisposed for some time due to suffering from gout, he was buried in the same site as his father by the prominent figures of Mazandaran, including the ispahbad Majd al-Din Dara, Sabiq al-Dawla Qazvini, Sayyid Hasim Alawi, Amir Surkhab. Ibn Isfandiyar, Muhammad ibn al-Hasan. An Abridged Translation of the History of Tabaristan, Compiled About A. H. 613. Trans. Edward G. Browne. Leyden: E. J. Brill. Madelung, W.. "The Minor Dynasties of Northern Iran". In Frye, R. N.. The Cambridge History of Iran, Volume 4: From the Arab Invasion to the Saljuqs. Cambridge: Cambridge University Press. Pp. 198–249. ISBN 978-0-521-20093-6. Bosworth, C. E.. "The Political and Dynastic History of the Iranian World".

In Frye, R. N.. The Cambridge History of Iran, Volume 5: The Saljuq and Mongol periods. Cambridge: Cambridge University Press. Pp. 1–202. ISBN 0-521-06936-X. Madelung, W.. "ĀL-E BĀVAND". Encyclopaedia Iranica, Vol. I, Fasc. 7. London u.a.: Routledge & Kegan Paul. Pp. 747–753. ISBN 90-04-08114-3. Babaie, Sussan. Persian Kingship and Architecture: Strategies of Power in Iran from the Achaemenids to the Pahlavis. I. B. Tauris. Pp. 1–288. ISBN 9780857734778

The video is themed around everyone being able to control YouTube Rewind, with various featured personalities describing what events they want to review. The video begins with actor Will Smith on the Swiss Alps suggesting the inclusion of popular video game Fortnite and YouTuber Marques Brownlee in the video; the camera cuts to Brownlee, other YouTubers and Twitch streamer Ninja conversing inside of a battle bus, a Fortnite reference. "I Like It" by famous rapper Cardi B is played on the radio during the scene as well. The following scene depicts a group of YouTube personalities surrounding a campfire. Casey Neistat and the Merrell Twins suggest that the Rewind should mention K-pop, after which the video cuts to Neistat, among others, imitating the music video of "Idol" by K-pop group BTS; the video cuts back to the campfire, as one YouTuber proposes a reference to the wedding of Prince Harry and Meghan Markle, but comedian Michael Dapaah establishes that the internet meme'Bongo Cat' will be the groom.

Following the wedding scene, Safiya Nygaard suggests a science experiment involving melting lipstick. Another suggests the inclusion of electronic musician Marshmello, whose mask is removed, revealing Mason Ramsey underneath; the video cuts to a group eating a mukbang in Korea. The scene shifts back to the campfire, when animator TheOdd1sOut suggests the inclusion of the "In My Feelings" challenge; the video cuts between scenes of various YouTubers and celebrities dancing to Drake's song "In My Feelings", including scenes of talk show hosts Trevor Noah and John Oliver performing dances from Fortnite. Here, animator Jaiden Animations included several easter eggs, comprising references to other memes and events of the year, such as Ugandan Knuckles, an invitation to Super Smash Bros. Ultimate and the KSI vs. Logan Paul boxing match, a group of items on the wall that spell out "Sub 2 Pewdiepie", as well as PewDiePie's swivel chair; the video once again cuts back to the group sitting around the campfire, with Lilly Singh claiming that the video should feature "the people who managed to do something bigger than themselves".