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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
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
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
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
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)
Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings A local complete intersection ring is
Complete_intersection_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. For Noetherian local rings, there is the following chain of inclusions: Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃
Regular_local_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
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
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 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
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
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
"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
International Congress of Mathematicians in 1966 (Moscow) The Kolchin Catenary Conjecture is a fundamental open problem in differential algebra related
Kolchin's_problems
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
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
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
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
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)
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
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
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
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, Missouri's tallest accessible structure
Gateway_Arch
Algebraic study of differential equations
differential ideal when characteristic sets identify both ideals. The Kolchin catenary conjecture states given a d > 0 {\textstyle d>0} dimensional irreducible
Differential_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
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
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
Line of thrust
architecture bearing weight. This includes aircraft, bridges, plus arches; see catenary arch. An arch won't collapse, when the line of thrust is entirely internal
Line_of_thrust
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)
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
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
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
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
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
Line on the Delhi Metro system
India's first operational ring metro. It is also known as the Ring Road Line, as it mostly passes alongside the city's Inner Ring Road. The Pink Line has
Pink_Line_(Delhi_Metro)
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
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
neighborhood was supposed to get a transit system, an 11.5 km (7.1 mi) catenary-free tramway, but these ideas have been scrapped. The Education City tram
Transport_in_Doha
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
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
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
Metro station in Delhi, India
February 2026. Retrieved 12 March 2026. "PM Modi Inaugurates India's First 'Ring Metro' In Delhi". NDTV. 8 March 2026. Retrieved 8 March 2026.{{cite news}}:
Nanaksar - Sonia Vihar metro station
Nanaksar_-_Sonia_Vihar_metro_station
Chinese high-speed train model
nearly 500 kilometers into Beijing. This caused a trip of the overhead catenary system in the section between Beijing South railway station and Beijing
China_Railway_CR300AF
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
Orbital rapid transit line under construction in Melbourne, Australia
freeways, some of which were constructed over the following decades. The M80 Ring Road through the outer western and northern suburbs was constructed in stages
Suburban_Rail_Loop
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
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)
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
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
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
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
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
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
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
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
College campus in Irvine, California, US
Administration. 2006. Archived from the original on January 26, 2008. "Catenary Maps". Catenary Maps. August 25, 2024. "University Pass". Orange County Transportation
Campus of the University of California, Irvine
Campus_of_the_University_of_California,_Irvine
Future line of the metro in Delhi, India
(4 ft 8+1⁄2 in) standard gauge Electrification 25 kV 50 Hz AC from overhead catenary Route map Legend Main Line Branch Line Magenta Line Terminal 1 IGI Airport
Golden_Line_(Delhi_Metro)
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
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
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
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
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
(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
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
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
Nagpur Metro's Aqua Line metro station
overhead catenary Services Preceding station Nagpur Metro Following station Dharampeth College towards Prajapati Nagar Aqua Line Rachana Ring Road Junction
Subhash Nagar metro station (Nagpur)
Subhash_Nagar_metro_station_(Nagpur)
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)
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)
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
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
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)
Transportation authority in Istanbul
The trolleybuses operated between Topkapı and Eminönü on a dual overhead catenary. 100 Trolleybuses operated on around 45 kilometres (28 mi) of route until
Istanbul Electricity, Tram and Tunnel Establishments
Istanbul_Electricity,_Tram_and_Tunnel_Establishments
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
Football stadium in Manchester, England
sweeping roof and support masts which are separate from the concrete bowl. A catenary cable is situated around the inner perimeter of the roof structure which
Etihad_Stadium
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
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
Superellipse Hurwitz surface Bowditch curve Brachistochrone Butterfly curve Catenary Clélies Cochleoid Cycloid Horopter Isochrone Isochrone of Huygens (Tautochrone)
List_of_mathematical_shapes
Landform within an ice field or glacier
Nunatak Cassandra Nunatak Castillo Nunatak Castor Nunatak Cat Nunatak Catenary Nunatak Cathedral Peak C. H. Ostenfeld Nunatak Chain Nunataks Cheeks Nunatak
Nunatak
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
Rapid transit system in India serving Delhi NCR
elevated line covers Delhi in a ring-shaped pattern. It is also known as the Ring Road Line, since it runs along the busy Ring Road. The line has interchanges
Delhi_Metro
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_wind_turbine
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
Nagpur Metro's Aqua Line metro station
22 October 2020; 5 years ago (2020-10-22) Electrified 25 kV 50 Hz AC overhead catenary Services Preceding station Nagpur Metro Following station Shankar Nagar
LAD_Square_metro_station
Delhi Metro station on pink line
record 22 travellators for seamless and efficient commuting. Situated on the Ring Road, near the junction of Benito Juarez Marg, this station is anticipated
Durgabai Deshmukh South Campus metro station
Durgabai_Deshmukh_South_Campus_metro_station
Tram network in the Netherlands
measuring 1,435 mm (4 ft 8+1⁄2 in), and are electrified using a 600 V DC catenary system, with some sections also capable of operating at 750 V DC for tram-train
Trams_in_The_Hague
Metro station in Delhi, India
Transport in Delhi Delhi Metro Rail Corporation Delhi Suburban Railway Inner Ring Road, Delhi Delhi Monorail Delhi Transport Corporation South Delhi New Delhi
Delhi Cantonment metro station
Delhi_Cantonment_metro_station
Railway line in Leipzig, Germany
1914. Due to the war, electrical operations were soon abandoned and the catenary was dismantled. Ownership of the whole line was transferred to Deutsche
Leipzig-Wahren–Leipzig-Engelsdorf railway
Leipzig-Wahren–Leipzig-Engelsdorf_railway
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
Metro station in Delhi, India
on 29 May 2018. It is electrified with 25 KW 50 Hz AC through overhead catenary. Its services are from preceding station Delhi Metro Following station
Dabri Mor - Janakpuri South metro station
Dabri_Mor_-_Janakpuri_South_metro_station
Architectural term for an arched roof
building an arch, a temporary support is needed while rings of voussoirs are constructed and the rings placed in position. Until the topmost voussoir, the
Vault_(architecture)
High-speed rail system of South Korea
bid for the supply of the core system, which included the rolling stock, catenary and signalling: consortia led by GEC-Alsthom, today Alstom, one of the
KTX
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 tunnel in western England
further accentuated with the 2015 lowering of the trackbed for electric catenary to be installed. The eastern portal at Corsham has a more modest brick
Box_Tunnel
Nagpur Metro's Aqua Line metro station
21 September 2023; 2 years ago (2023-09-21) Electrified 25 kV 50 Hz AC overhead catenary Services Preceding station Nagpur Metro Following station Nagpur Railway
Cotton_Market_metro_station
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