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In mathematics, a commutative ring R is catenary if for any pair of prime ideals p, q, any two strictly increasing chains p = p0 ⊂ p1 ⊂ ... ⊂ pn = q of
Catenary_ring
Noetherian ring in algebra
#formally catenary ring below.) Precisely, a quasi-unmixed ring is a ring in which the unmixed theorem, which characterizes a Cohen–Macaulay ring, holds
Quasi-unmixed_ring
Curve formed by a hanging chain
catenoid—the shape assumed by a soap film bounded by two parallel circular rings. The catenary is also called the alysoid, chainette, or, particularly in the materials
Catenary
Topics referred to by the same term
Catenary arch, an arch that follows a catenary curve Catenary ring, a type of mathematical ring Overhead catenary, another name for the overhead line system
Catenary_(disambiguation)
Type of commutative ring in mathematics
rings, there is the following chain of inclusions. Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings
Cohen–Macaulay_ring
Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings A local complete intersection ring is
Complete_intersection_ring
Local ring in commutative algebra
rings, there is the following chain of inclusions. Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings
Gorenstein_ring
Mathematical ring with well-behaved ideals
universally catenary. Goldie's theorem Given a ring, there is a close connection between the behaviors of injective modules over the ring and whether the ring is
Noetherian_ring
Type of ring in commutative algebra
rings, there is the following chain of inclusions: Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings
Regular_local_ring
Concept in mathematics
the generic freeness plays a main role and the notion of "universally catenary ring" is a key in the proof, see Eisenbud, Ch. 14 of "Commutative algebra
Morphism of algebraic varieties
Morphism_of_algebraic_varieties
Branch of algebra
Practically all noetherian rings that appear in applications are catenary. Ratliff proved that a noetherian local integral domain R is catenary if and only if for
Ring_theory
Algebraic structure
properties of regular rings (such as the property of being universally catenary rings, which means that the (co)dimension of primes is well-behaved), but
Commutative_ring
Concept in commutative algebra
universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur
Excellent_ring
In mathematics, dimension of a ring
length. A ring is called universally catenary if any finitely generated algebra over it is catenary. Nagata gave an example of a Noetherian ring which is
Krull_dimension
"Canonical module" is an alternative term for a dualizing module. catenary A ring is called catenary if for every two prime ideals, one contained in the other
Glossary of commutative algebra
Glossary_of_commutative_algebra
semichain modules. The notion of a catenary ring has "chain" as its namesake, but it is in general not related to chain rings. In the 1930s, Gottfried Köthe
Serial_module
Kurosh conjecture, the Nullstellensatz and the catenary property for prime ideals. If R is a PI-ring and K is a subring of its center such that R is
Polynomial_identity_ring
International Congress of Mathematicians in 1966 (Moscow) The Kolchin Catenary Conjecture is a fundamental open problem in differential algebra related
Kolchin's_problems
Surface of revolution of a catenary
In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning
Catenoid
Type of ring in commutative algebra
quasi-excellent ring, and if in addition it is universally catenary it is called an excellent ring. A (Noetherian) ring R containing a field k is called geometrically
G-ring
Rail line
but the catenary is still missing. Between Pichling and Ebelsberg a large section of the overgrown tracks has been removed to build a ring road and has
Florianerbahn
Railway line in Germany
The Nuremberg Ring Railway is the ring railway for freight that runs at a distance of three to four kilometres (2 to 2+1⁄2 mi) from the center of Nuremberg
Nuremberg_Ring_Railway
Catalan architect (1852–1926)
stable, covered longeing ring and porter's lodge: the stable has a rectangular base and catenary arches; the longeing ring has a square base with a hyperboloid
Antoni_Gaudí
Railway line in Germany
Hennigsdorf (now owned by Bombardier) were used on the line. In 1973, the catenary was removed, although some masts were preserved and reused for the 15 kV
Berlin_outer_ring
US National Historic Landmark in St. Louis, Missouri
United States. Clad in stainless steel and built in the form of a weighted catenary arch, it is the world's tallest arch and Missouri's tallest accessible
Gateway_Arch
electric catenary in 1914. This was one of the first electrified lines in Germany. Due to the two world wars and the resulting dismantling of the catenary in
Leipzig_Freight_Ring
Railway line in Germany
S47 use a section of the southern and western ring, while lines S8 and S85 use sections of the eastern ring. The combined number of passengers is about
Berlin_Ringbahn
Structure on which road signs are mounted
in Georgia Butterfly gantry near Raubling, Germany On the Jakarta Outer Ring Road Gantries placed on either side of the carriageway on the A4 autoroute
Gantry_(transport)
Rail line in Bavaria, Germany
Munich North Ring (German: Münchner Nordring), section of which are only used by freight trains, is a railway bypass on the northern edge of the Bavarian
Munich_North_Ring
Algebraic study of differential equations
differential ideal when characteristic sets identify both ideals. The Kolchin catenary conjecture states given a d>0 dimensional irreducible differential algebraic
Differential_algebra
Study of dimension in algebraic geometry
universally catenary. This implies for example that a polynomial ring k [ x 1 , … , x d ] {\displaystyle k[x_{1},\dots ,x_{d}]} is universally catenary since
Dimension_theory_(algebra)
Railroad power system
transformers. Renew 40 miles (64 km) of catenary in Maryland. Renew 18 miles (29 km) of catenary in Pennsylvania. Continue catenary renewal along Hell Gate Line
Amtrak's 25 Hz traction power system
Amtrak's_25_Hz_traction_power_system
Building by Antoni Gaudí in Barcelona, Spain
influence through the use of white on the walls. It contains a series of sixty catenary arches that creates a space which represents the ribcage of an animal.
Casa_Batlló
locally noetherian, dimension, catenary, Gorenstein. local complete intersection The local rings are complete intersection rings. See also: regular embedding
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Japanese mathematician
rings contains several other counterexamples he found, such as a commutative Noetherian ring that is not catenary, and a commutative Noetherian ring of
Masayoshi_Nagata
Rapid transit station in Boston, Massachusetts, US
between Airport station and Wood Island station around 1999–2000. Overhead catenary was re-installed between Maverick and Airport stations as part of the work
Airport_station_(MBTA)
are DC1500V overhead wires. The overhead wires is divided into overhead catenary and overhead conductor rails. Except for the earlier construction of lines
Shanghai_Metro_rolling_stock
Railway line in Beijing, China
The Beijing Northeast Ring railway (Chinese: 北京东北环线; pinyin: Běijīng dōngběi huánxiàn), also known as Shuangqiao–Shahe railway or Shahe railway, is a link
Beijing Northeast Ring railway
Beijing_Northeast_Ring_railway
Rail line in Beijing, China
a large comprehensive railway test base located near the Northeast Fifth Ring Road in Chaoyang District, Beijing, China. It is currently the only circular
Beijing National Railway Test Centre
Beijing_National_Railway_Test_Centre
Line on the Delhi Metro system
first operational ring metro. It is also informally known as the Ring Road Line, as it mostly passes alongside the city's Inner Ring Road. The Pink Line
Pink_Line_(Delhi_Metro)
Tram system in Antwerp, Belgium
subway rail setup — 1,000 mm (3 ft 3+3⁄8 in) track gauge and a 600 volts catenary power feed. The tram network is connected to the Flemish Tram and Bus Museum
Antwerp_tramway_network
Nagpur Metro's Aqua Line metro station
Rachana Ring Road Junction is an elevated metro station on the East-West corridor of the Aqua Line of Nagpur Metro in Nagpur, India. It was opened on 9
Rachana Ring Road Junction metro station
Rachana_Ring_Road_Junction_metro_station
Museum in Manhattan, New York
for a special tricycle with square wheels, which operates smoothly on a catenary surface. In 2006 the Goudreau Museum on Long Island, at the time the only
National Museum of Mathematics
National_Museum_of_Mathematics
Metro station in Delhi, India
in Delhi List of Delhi Metro stations "PM Modi Inaugurates India's First 'Ring Metro' In Delhi". NDTV. 8 March 2026. Retrieved 8 March 2026.{{cite news}}:
Khajuri_Khas_metro_station
Metro station in Delhi, India
Transport Corporation Transport in Delhi "PM Modi Inaugurates India's First 'Ring Metro' In Delhi". NDTV. 8 March 2026. Retrieved 8 March 2026.{{cite news}}:
Burari_metro_station
Railway station in Leipzig, Germany
Retrieved 22 August 2021. Since 1915 catenaries were dismantled to gain cupper for war production. Since 1946 catenaries and overhead line masts were dismantled
Leipzig-Thekla_station
Metro Station in Delhi, India
Deepali Chowk–Majlis Park stretch nears launch; big traffic relief on outer ring road". The Times of India. 22 February 2026. Archived from the original on
Deepali_Chowk_metro_station
Rail line in Bavaria, Germany
The Erding ring closure (German: Erdinger Ringschluss) is a project to build a 30.2-kilometre (18.8 mi) long, double-track railway that is planned to run
Erding_Ring_Closure
High-speed rail project in England
"Change of HS2 railway specification and scope" (PDF). GOV.UK. "SNCF catenary design adopted as HS2 begins electrification procurement". Railway Gazette
High_Speed_2
(algebraic) Butterfly curve (transcendental) Butterfly lemma Butterfly graph Catenary, derived from the Latin word catēna, which means "chain". Cissoid (from
Glossary of shapes with metaphorical names
Glossary_of_shapes_with_metaphorical_names
Railway line in Germany
has been closed, the Golm–Priort section is now a part of the Berlin outer ring and has been rebuilt as a double-track main line. At the turn of the 20th
Jüterbog–Nauen_railway
Chinese high-speed train model
railway station. On April 30, 2021, the CR300AF began operating on the Hainan Ring High-Speed Railway. The CR300AF features a new front end design, with a shorter
China_Railway_CR300AF
Neighbourhood of Delhi in North East Delhi, India
IV. The corridor adds eight new stations and will complete India’s first ring-shaped metro, doubling the system’s operational efficiency. Corridor and
Sonia_Vihar
Line of Wuhan Metro
Caidian District. Line 6 is the first Wuhan Metro line to utilize overhead catenary instead of third rail for electrification. 武汉两条地铁线今日开门迎客 6号线堪称超级换乘线_大楚网_腾讯网
Line_6_(Wuhan_Metro)
Rapid transit in Pune
(4 ft 8+1⁄2 in) standard gauge Electrification 25 kV 50 Hz AC overhead catenary (Purple and Aqua Line) 750 V DC Third rail (Pink Line) Average speed 40 km/h
Pune_Metro
Aspect of architectural history
calculus and the study of statics. There were numerous studies on the catenary curve as the most stable form for arches and vaults and the ideal shape
18th-century_Western_domes
Type of wind turbine
is the catenary, which uses free hanging chain mooring lines, with the station-keeping force provided by the self-weight of the chain. A catenary mooring
Floating_offshore_wind
Metro station in Delhi, India
Metro. Soorghat was the only station that was being skipped past from the ring network operation due to the delayed construction and finalization of the
Soorghat_metro_station
Curved structure that spans a space and may support a load
weighted catenary) Palau Güell: Elliptical Three parabolic-looking curves in particular are of significance to the arch design: parabola itself, catenary, and
Arch
Romanian electric locomotive
uneconomical to operate, not to mention dangerous for the locomotive and the catenary. The unit, EB 001, was rebuilt and renumbered into EA 931. The air brakes
Electroputere_LE5100
Metro station in Delhi, India
Transport Corporation Transport in Delhi "PM Modi Inaugurates India's First 'Ring Metro' In Delhi". NDTV. 8 March 2026. Retrieved 8 March 2026.{{cite news}}:
Jagatpur - Wazirabad metro station
Jagatpur_-_Wazirabad_metro_station
Anglican cathedral in London, England
extent than had been done by Michelangelo at St Peter's, drafting both as catenary curves, rather than as hemispheres. Between the inner and outer domes,
St_Paul's_Cathedral
Temporary structure used to support a work crew and materials
suspension scaffolds, and masons’ multipoint adjustable suspension scaffolds; catenary; float (ship); interior hung; needle beam; multilevel suspended; mobile;
Scaffolding
Railway line in Germany
single-track electrified line links the Stadtbahn with the Berlin outer ring (German: Berliner Außenring, BAR). Following the closure of the old Berlin
VnK_Railway
Device used to secure a vessel to the seabed
arrangements are used for some types of single buoy moorings, like the catenary anchor leg mooring (CALM) used for loading and unloading liquid cargoes
Anchor
Line on the Delhi Metro system
676 mm (5 ft 6 in) broad gauge Electrification 25 kV 50 Hz AC from overhead catenary Route map Legend Noida Electronic City Noida Sector 62 Noida Sector 59
Blue_Line_(Delhi_Metro)
Rapid transit system in Jakarta, Indonesia
Like the KRL Commuterline, the MRT trains are also powered via overhead catenary. MRT Jakarta officially call the trains Ratanggas (singular Ratangga),
Jakarta_MRT
Building complex designed by Antoni Gaudí
longeing ring and gatehouses. The stable is rectangular and roofed with a high Catalan vault adopting a catenary curve; the longeing ring has a square
Güell_Pavilions
Approved Metro Line in India
Junction, through Roda Mistry college lane, Khajaguda Junction, touching Outer Ring Road at Nanakramguda junction and traverse along ORR to Shamshabad Airport
Hyderabad Airport Metro Express
Hyderabad_Airport_Metro_Express
Bridge in New York City
the engineers on site, but also understood mathematics, calculations of catenary curves, strengths of materials, bridge specifications, and the intricacies
Brooklyn_Bridge
Rapid transit line in Ho Chi Minh City, Vietnam
of Suoi Tien Terminal station - Mỹ Phước – Tân Vạn Road (coincided with Ring Road 3 until Bình Chuẩn Bridge) - Provincial Road 742 - Bình Dương New City
Line 1 (Ho Chi Minh City Metro)
Line_1_(Ho_Chi_Minh_City_Metro)
used in partial pressure gas blending. catenary anchor leg mooring Single point mooring type named for the catenary curve of the anchor cables that hold
Glossary of underwater diving terminology: A–C
Glossary_of_underwater_diving_terminology:_A–C
Train station of the Delhi Metro
station is located on the Pink Line of the Delhi Metro. It is one of the ring completion network station of Pink Line, and is the part of Delhi Metro's
Jharoda_Majra_metro_station
Metro station in Delhi, India
Delhi Metro. The station was opened on 6 August 2018, and is situated on the Ring Road in Bhikaji Cama Place, which is named after Bhikaji Cama, an Indian
Bhikaji Cama Place metro station
Bhikaji_Cama_Place_metro_station
Metro station in Delhi, India
months ago (5 January 2025) Electrified 25 kV 50 Hz AC through overhead catenary Passengers Jan 2025 21,262/day 659,120/ Month average Services Preceding
Krishna Park Extension metro station
Krishna_Park_Extension_metro_station
Planned high-speed rail line
Electrification 25 kV AC, 50 Hz, overhead catenary Operating speed 350 km/h (220 mph) Route map v t e Legend Chennai Chennai Ring Road Tamil Nadu Andhra Pradesh
Chennai–Hyderabad high-speed rail corridor
Chennai–Hyderabad_high-speed_rail_corridor
Largest rail underpass in Spain
meters per day. The tunnel was lined with prefabricated cement voissoirs, ring-shaped supports, that was manufactured used some the excavated material onsite
Guadarrama_Tunnel
Nagpur Metro's Orange Line metro station
8 March 2019; 7 years ago (2019-03-08) Electrified 25 kV 50 Hz AC overhead catenary Services Preceding station Nagpur Metro Following station Ujjwal Nagar
Airport metro station (Nagpur)
Airport_metro_station_(Nagpur)
single mast mounted close to the bow and only one sail, usually on a gaff. catenary The curve of a deployed anchor chain. catharpin A short rope or iron clamp
Glossary of nautical terms (A–L)
Glossary_of_nautical_terms_(A–L)
Metro station in Delhi, India
As a part of the Sarai Kale Khan multi-model transport hub on the Inner Ring Road, the Sarai Kale Khan – Nizamuddin metro station facilitates the transfer
Sarai Kale Khan – Nizamuddin metro station
Sarai_Kale_Khan_–_Nizamuddin_metro_station
Transit line in Chennai, India
Government of India for final approval. A depot near Pattabiram stretch of Outer Ring Road is also planned as a part of the expansion. § under construction * -
Red_Line_(Chennai_Metro)
Transit line in Chennai, India
section reserved for women. The trains operate on a 25 kV AC traction catenary system. The trains are connected to the grid via overhead electric cables
Green_Line_(Chennai_Metro)
Railway line in Germany
(4 ft 8+1⁄2 in) standard gauge Electrification 15 kV/16.7 Hz AC overhead catenary Operating speed 120 km/h (74.6 mph) (max) with 2 sections of 100 km/h (62
Munich East–Deisenhofen railway
Munich_East–Deisenhofen_railway
Above-ground structure for bulk transfer and distribution of electricity
Flexible conductors supported at their ends approximate the form of a catenary, and much of the analysis for construction of transmission lines relies
Overhead_power_line
Street-running rail transit
this ceased to be a problem. In the 2000s, several companies introduced catenary-free designs: Alstom's Citadis line uses a third rail, Bombardier's PRIMOVE
Tram
All Latin and Greek roots beginning with G
catarrhine, catastrophe, catatonia, cathode, cation caten- chain Latin catena catenary, concatenation cathar- pure Greek καθαρός (katharós) catharsis, cathartic
List of Greek and Latin roots in English/A–G
List_of_Greek_and_Latin_roots_in_English/A–G
Railway line in Beijing, China
metres (112 ft) underground. The line is the first line to 1500 V overhead catenary as opposed to 750 V third rail used on all earlier lines. Some of the station
Line_6_(Beijing_Subway)
Rapid transit system in Ho Chi Minh City, Vietnam
7 km (380.7 mi) (planned) Track gauge 1,435 mm (4 ft 8+1⁄2 in) standard gauge Electrification Overhead catenary Top speed 110 km/h (68 mph) System map
Ho_Chi_Minh_City_Metro
Metro station in Noida, India
9 March 2019 (2019-03-09) Electrified Single phase 25 kV 50 Hz AC through overhead catenary Services Preceding station Delhi Metro Following station Noida Sector 62
Noida Electronic City metro station
Noida_Electronic_City_metro_station
2,209,028 The Gateway Arch is a 630-foot (192 m) (both high and wide) catenary arch built in the 1960s to commemorate the Lewis and Clark Expedition,
List of national parks of the United States
List_of_national_parks_of_the_United_States
Dutch mathematician and physicist (1629–1695)
as Galileo thought. Huygens would later label that curve the catenaria (catenary) in 1690 while corresponding with Gottfried Leibniz. In the next two years
Christiaan_Huygens
Commuter rail system Between Yogyakarta and Surakarta, Indonesia
and support tourism in Yogyakarta–Surakarta region. In 2016, overhead catenary poles began to be stacked at Solo Jebres Station; since then the construction
KAI_Commuter_Yogyakarta_Line
Transit line in Chennai, India
dedicated section reserved for women. The trains operate on 25 kV AC traction catenary system. The trains are connected to the grid via overhead electric cables
Blue_Line_(Chennai_Metro)
Viaduct in Greater Manchester, England
by the London and North Western Railway (LNWR). In the 1960s, overhead catenary lines were installed by British Rail for the West Coast Main Line electrification
Stockport_Viaduct
Railway line of China
speed record of 487.3 km/h (302.8 mph) during a test run. The overhead catenary work was completed on February 4, 2011 for the entire line. According to
Beijing–Shanghai high-speed railway
Beijing–Shanghai_high-speed_railway
Town in San Mateo County, California
more efficient diesel trains" should supplant plans for "century-old catenary electrical line technology." Atherton mayor Rick De Golia was quoted as
Atherton,_California
United States naval transmission facility near Oso, Washington
efficiently radiate the VLF waves. The antenna, shown above, consists of ten catenary cables, 5,640–8,700 ft (1,719–2,652 m, 1.1–1.6 miles) long, suspended in
Jim_Creek_Naval_Radio_Station
Metro line of the Mumbai Metro
is electrified at 25 kV 50 Hz AC, with power provided via an overhead catenary. The MMRDA signed an agreement with Adani Electricity Mumbai Limited to
Yellow_Line_(Mumbai_Metro)
Heatwaves in Europe in 2026
hitting the vulnerable the hardest. Heat-induced bending of rail tracks and catenary wires prompted many cancellations, particularly in France and Belgium.
2026_European_heatwaves
Rapid transit system in Beijing, China
the suburbs beyond the 5th Ring Road. Lines 15, S1 along with the Changping, Daxing, Yanfang lines extend beyond the 6th Ring Road. Line 11 currently runs
Beijing_Subway
travel, tourism, insurance
CATENARY RING
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travel, tourism, insurance