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Analogue of a prime number in a commutative ring
specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and
Prime_element
In algebra, element without non-trivial factors
ideal generated by a prime element is a prime ideal, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal
Irreducible_element
1976 studio album by Elvin Jones
The Prime Element is a jazz album by jazz drummer Elvin Jones, originally released in 1976 as part of the "Blue Note Re-issue Series". The pieces compiled
The_Prime_Element
Branch of number theory
general, a prime element is an element of the ring satisfying the key property of prime numbers given by Euclid's lemma: a prime element is an element p of
Algebraic_number_theory
Ideal in a ring which has properties similar to prime elements
commutative ring R is prime if it has the following two properties: If a and b are two elements of R such that their product ab is an element of P, then a is
Prime_ideal
Number divisible only by 1 and itself
ring elements. Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an
Prime_number
Commutative ring with no zero divisors other than zero
which the zero ideal {0} is a prime ideal. An integral domain is a nonzero commutative ring for which every nonzero element is cancellable under multiplication
Integral_domain
About products of primitive polynomials
given below for the more general case. Note that an irreducible element of Z (a prime number) is still irreducible when viewed as constant polynomial
Gauss's_lemma_(polynomials)
Type of integral domain
irreducible element is prime. (In any integral domain, every prime element is irreducible, but the converse does not always hold. For instance, the element z ∈
Unique_factorization_domain
Set whose pairs have minima and maxima
completely join prime. The dual notion is meet prime. Every join-prime element is also join irreducible, and every meet-prime element is also meet irreducible
Lattice_(order)
Concept in abstract algebra
domain with a unique non-zero prime ideal. R {\displaystyle R} is a principal ideal domain with a unique irreducible element (up to multiplication by units)
Discrete_valuation_ring
In mathematics, an element p of a partial order (P, ≤) is a meet prime element when p is the principal element of a principal prime ideal. Equivalently
Prime_(order_theory)
Mathematical structure where elements are primal
an irreducible element is primal if and only if it is a prime element. Consequently, in a pre-Schreier domain, every irreducible is prime. In particular
Schreier_domain
Topics referred to by the same term
two nontrivial 3-manifolds Prime element, in algebra Prime form of a Riemann surface Prime ideal, a subset of a ring Prime knot, a knot that cannot be
Prime_(disambiguation)
Earth, water, air, fire, and (later) aether
first to use the term "element (στοιχεῖον, stoicheîon)" in reference to air, fire, earth, and water. The ancient Greek word for element, stoicheion (from stoicheo
Classical_element
decomposition of an element into irreducibles need not be unique; stated differently, an irreducible element is not necessarily a prime element. Important examples
Atomic_domain
numbers) shows. Every completely join-prime element of a lattice is compact.[citation needed] A poset in which every element is the supremum of the directed
Compact_element
Relationship between certain categories
is a meet-prime element. Now the set-inverse of p −1(0) given by p −1(1) is a completely prime filter because p −1(0) is a principal prime ideal. It turns
Stone_duality
Natural number
number following 1 and preceding 3. It is the smallest and the only even prime number. Because it forms the basis of a duality, it has religious and spiritual
2
Algebraic structure
of powers of prime numbers. It is also known as the fundamental theorem of arithmetic. An element a {\displaystyle a} is a prime element if whenever a
Commutative_ring
Polynomial without nontrivial factorization
an element f of R that is neither zero nor a unit is called irreducible if there are no non-units g and h with f = gh. One can show that every prime element
Irreducible_polynomial
Song by Kendrick Lamar
"Element" (stylized as "ELEMENT.") is a song by American rapper Kendrick Lamar, from his fourth studio album Damn, released on April 14, 2017. The fourth
Element_(song)
Political officer in Nazi Germany
activities to the local Gestapo office; thus the Block Warden was the prime element of the Nazi totalitarian state. This was helped by keeping files on
Blockleiter
Merry-Go-Round (Blue Note, 1971) – piano Mr. Jones (Blue Note, 1972) – piano The Prime Element (Blue Note, 1973) – keyboards On the Mountain (PM, 1975) – keyboards
Jan_Hammer_discography
Mathematical group that can be generated as the set of powers of a single element
groups of prime order. The cyclic groups of prime order are thus among the building blocks from which all groups can be built. For any element g in any
Cyclic_group
Cost of the materials in a product
manufacturing. The manufacture of products or goods requires material as the prime element. In general, these materials are divided into two categories. These
Direct_materials_cost
Condition under which an odd prime is a sum of two squares
integers in the quadratic field, then an odd prime number p, not dividing d, is either a prime element in O d , {\displaystyle {\mathcal {O}}_{\sqrt
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
defined by the relation xa = σ(a)x ∀a ∈ R. prime 1. An element x of an integral domain is a prime element if it is not zero and not a unit and whenever
Glossary_of_ring_theory
1794 novel by Ann Radcliffe
that an incident "is in its very realism as much charged with that prime element of Radcliffian romance, the mysterious, as any that the ingenuity of
The_Mysteries_of_Udolpho
(infimum, meet) Limit superior and limit inferior Irreducible element Prime element Compact element Cofinal and coinitial set, sometimes also called dense Meet-dense
List_of_order_theory_topics
American jazz trumpeter (1938–1972)
Moments; Lonnie Smith's Think and Turning Point; Elvin Jones's The Prime Element; Jack Wilson's Easterly Winds; Reuben Wilson's Love Bug; Larry Young's
Lee_Morgan
Set with associative invertible operation
third element within the same set and the following conditions must hold: the operation is associative, it has an identity element, and every element of
Group_(mathematics)
Function used in local class field theory related to reciprocity laws
number field containing the nth roots of unity, p is a prime ideal not dividing n, π is a prime element of the local field of p, and a is coprime to p, then
Hilbert_symbol
On prime factors of integer products
algorithm Fundamental theorem of arithmetic Irreducible element Prime element Prime number Prime ideal It is also called Euclid's first theorem although
Euclid's_lemma
Equivalence of distributive lattices and set families
a join-prime element must be join-irreducible. Equivalently, an element that is not join-irreducible is not join-prime. For, if an element x is not
Birkhoff's representation theorem
Birkhoff's_representation_theorem
regular ring. prime 1. A prime ideal is a proper ideal whose complement is closed under multiplication. 2. A prime element of a ring is an element that generates
Glossary of commutative algebra
Glossary_of_commutative_algebra
1994 single by Jann Arden
original on November 20, 2008. LeBlanc, Larry (April 13, 1996). "Work Is Prime Element of Success for A&M's Arden". Billboard. Vol. 108, no. 15. p. 20. "Top
Insensitive_(song)
2007 video game
Alpha Prime is a first-person shooter video game developed by Black Element Software and released in 2006–2007. The script for the game's story was written
Alpha_Prime
Transformers character
Optimus Prime, also known in Japan as Convoy (コンボイ), is a fictional character and the protagonist of the Transformers franchise. Generally depicted as
Optimus_Prime
Any one of the distinct objects that make up a set in set theory
In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called A containing
Element_of_a_set
American professor of the history of technology
Technology comes in packages, big and small. Although technology might be a prime element in many public issues, nontechnical factors take precedence in technology-policy
Melvin_Kranzberg
Any of the fifteen lanthanides plus scandium and yttrium
plentiful in the entire Earth's crust (cerium being the 25th-most-abundant element at 68 parts per million, more abundant than copper), but in practice they
Rare-earth_element
Statement in abstract algebra
and some submodule F {\displaystyle F} of M {\displaystyle M} . For a prime element p in R we can then speak of N p = { m ∈ t M ∣ ∃ i , m p i = 0 } {\displaystyle
Structure theorem for finitely generated modules over a principal ideal domain
Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
DC Comics character
Superboy-Prime and the Legion of Super-Villains. The pre-Crisis Brainiac 5 uses a special lightning rod to enhance the transmutation ability of Element Lad's
Element_Lad
American R&B/soul jazz guitarist (1942–2011)
Elvin Jones, At This Point in Time (Blue Note, 1998) Elvin Jones, The Prime Element (Blue Note, 1976) Salena Jones, My Love (JVC, 1981) Kimiko Kasai, This
Cornell_Dupree
Faction of sentient robots from the Transformers universe
each imbued with a unique "life force" known as a "spark." Led by Optimus Prime in most stories, the Autobots believe that "freedom is the right of all
Autobot
Algebraic structure
case where q {\displaystyle q} is prime is Fermat's little theorem. If a {\displaystyle a} is a primitive element in G F ( q ) {\displaystyle \mathrm
Finite_field
Prime Minister of the United Kingdom from 2019 to 2022
Johnson (born 19 June 1964) is a British politician and writer who served as Prime Minister of the United Kingdom and Leader of the Conservative Party from
Boris_Johnson
Argentine-American composer (born 1948)
nominee Remembrances with soloist Jon Faddis, and many others. 1976: The Prime Element 1977: Graffiti 1979: Galaxy Dust 1983: Prometheus 1984: New York Toccata
Carlos_Franzetti
Generator of the multiplicative group of a finite field
GF(q); this means that each non-zero element of GF(q) can be written as αi for some natural number i. If q is a prime number, the elements of GF(q) can be
Primitive element (finite field)
Primitive_element_(finite_field)
Element in a ring whose some power is 0
called a reduced ring. Every nilpotent element x {\displaystyle x} in a commutative ring is contained in every prime ideal p {\displaystyle {\mathfrak {p}}}
Nilpotent
1997 video game
motivation of the characters of Criticom) turns out to be the Core Prime Element of a Master Key, one which holds the power to all the secrets in the
Dark_Rift
Mathematical element
In commutative algebra, an element b of a commutative ring B is said to be integral over a subring A of B if b is a root of some monic polynomial over
Integral_element
Rational numbers with root 5 added
action of the group of units. A prime element of a ring, analogous to a prime number among the integers, is an element γ {\displaystyle \gamma } such
Golden_field
Aspect of algebraic number theory
Frobenius element of an unramified prime P does not depend on which Pj we take. Furthermore, in the abelian case, associating an unramified prime of K to
Splitting of prime ideals in Galois extensions
Splitting_of_prime_ideals_in_Galois_extensions
Chemical element with atomic number 3 (Li)
Lithium (from Ancient Greek: λίθος, líthos, 'stone') is a chemical element; it has symbol Li and atomic number 3. It is a soft, silvery-white alkali metal
Lithium
American jazz bassist
the Village Vanguard (Enja, 1968) Poly-Currents (Blue Note, 1969) The Prime Element (Blue Note, 1969) Coalition (Blue Note, 1970) With J. J. Johnson J Is
Wilbur_Little
Natural number
multiplied by 1 equals the same number. 1 is by convention not considered a prime number. In digital technology, 1 represents the "on" state in binary code
1
Mathematical formal group law
ideal to Frobenius and the derivative at the origin is the prime element. For each element a in Zp there is a unique endomorphism f of the Lubin–Tate
Lubin–Tate_formal_group_law
Natural number
prime and the first good prime. 11 forms the first pair of sexy primes with 5. 5 is the second Fermat prime, of a total of five known Fermat primes.
5
American television personality (born 1968)
"face of the network", bringing an "element of rowdy, mass-market culture to American food television" and that his "prime-time shows attract more male viewers
Guy_Fieri
Existence of group elements of prime order
is a finite group and p is a prime number dividing the order of G (the number of elements in G), then G contains an element of order p. That is, there is
Cauchy's theorem (group theory)
Cauchy's_theorem_(group_theory)
American economist and statistician (1912–2006)
Estonia from an impoverished Soviet republic to the "Baltic Tiger". A prime element of Laar's program was introduction of the flat tax. Laar won the 2006
Milton_Friedman
Algebraic invariant
{u}}_{2})\cdots l({\overline {u}}_{n})} where π {\displaystyle \pi } a prime element, meaning Ord v ( π ) = 1 {\displaystyle {\text{Ord}}_{v}(\pi )=1} ,
Milnor_K-theory
1998 studio album by Elvin Jones
Foster) – 7:34 "Currents/Pollen" (Gene Perla, Don Garcia) – 11:13 "The Prime Element" (Omar Clay) – 8:19 "Whims of Bal" (Clay) – 12:23 "Pauke Tanz" (Clay)
At_This_Point_in_Time
Abstract algebra concept
above definition, prime ring may also refer to the minimal non-zero subring of a field, which is generated by its identity element 1, and determined by
Prime_ring
mathematics, an element a of a commutative ring R is called (relatively) prime to an ideal I if whenever ab is an element of I then b is also an element of I. A
Primal_ideal
Characterization of how many integers are prime
{\displaystyle \log _{e}(x)} . In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of prime numbers among the positive integers. It
Prime_number_theorem
Nonempty, upper-bounded, downward-closed subset
some element of A is also in I. So this is just a specific prime ideal that extends the above conditions to infinite meets. The existence of prime ideals
Ideal_(order_theory)
Type of prime number
5={\frac {3+7}{2}}} , no element of a twin prime pair is a balanced prime. This follows from the fact that there are no twin prime triplets of the form (
Balanced_prime
One of the four classical elements
least volume with the greatest surface area. This also makes fire the element with the smallest number of sides, and Plato regarded it as appropriate
Fire_(classical_element)
Decomposition of a number into a product
until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem
Integer_factorization
Comics character
victim of the military–industrial complex, of which Kord's company is a prime element. Utlilizing powered armor and a stylized weapon that hurls explosives
Overthrow_(comics)
As simple a model as possible, in model theory
particular model theory, a prime model is a model that is as simple as possible. Specifically, a model P {\displaystyle P} is prime if it admits an elementary
Prime_model
Type of Gödel numbering in mathematics
{\displaystyle p\mid (i-j)\cdot m} . Since p is a prime element (note that the irreducible element property is used), we get p ∣ i − j ∨ p ∣ m {\displaystyle
Gödel_numbering_for_sequences
Numbers obtained by adding the two previous ones
In mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the
Fibonacci_sequence
Prime Minister of the United Kingdom from 2010 to 2016
Chipping Norton (born 9 October 1966), is a British politician who served as Prime Minister of the United Kingdom from 2010 to 2016. Until 2015, he led the
David_Cameron
Map raising elements to the pth power, in characteristic p
commutative rings with prime characteristic p, an important class that includes finite fields. The endomorphism maps every element to its pth power. In
Frobenius_endomorphism
Infinite sequence of prime numbers
Euclid–Mullin sequence is an infinite sequence of distinct prime numbers, in which each element is the least prime factor of one plus the product of all earlier elements
Euclid–Mullin_sequence
Smallest integer n for which n equals 0 in a ring
only a single element 0. If a nontrivial ring R does not have any nontrivial zero divisors, then its characteristic is either 0 or prime. In particular
Characteristic_(algebra)
Period of the Government of the United Kingdom since 2024
The premiership of Keir Starmer as Prime Minister of the United Kingdom began on 5 July 2024 when he accepted an invitation from King Charles III to form
Premiership_of_Keir_Starmer
Chemical element with atomic number 94 (Pu)
Plutonium is a chemical element; it has symbol Pu and atomic number 94. It is a silvery-gray actinide metal that tarnishes when exposed to air, and forms
Plutonium
Result concerning ideals of commutative rings
In algebra, the prime avoidance lemma says that if an ideal I in a commutative ring R is contained in a union of finitely many prime ideals Pi's, then
Prime_avoidance_lemma
1960 bossa nova jazz standard
Carlos Jobim – The Composer of Desafinado Plays (1963) Elvin Jones – The Prime Element (rec. 1969 – released in 1976) Karin Krog/Jacob Young – Where Flamingo
Once_I_Loved
Commutative ring with a well behaved theory of prime factorization
{\displaystyle A} ), any nonzero element of A {\displaystyle A} is contained in only a finite number of height 1 prime ideals. It is also possible to characterize
Krull_ring
Episode list for an animated series
elements, there are four nations that correspond with each element. Not everyone can bend an element, and those that can only bend one. However, the Avatar
List of Avatar: The Last Airbender episodes
List_of_Avatar:_The_Last_Airbender_episodes
Field extension generated by a one element
a single element, called a primitive element. Simple extensions are well understood and can be completely classified. The primitive element theorem states
Simple_extension
Ancient algorithm for generating prime numbers
next prime, and all the numbers in it below the square of its first element will be prime too. Thus, when generating a bounded sequence of primes, when
Sieve_of_Eratosthenes
Metroid Prime is the titular main antagonist and final boss of Metroid Prime. Its later form, Dark Samus, is a major antagonist in Metroid Prime 2: Echoes
List_of_Metroid_characters
Theorem in number theory
when n is prime was proved by Furtwangler in 1902.[citation needed] The Hasse norm theorem can be deduced from the theorem that an element of the Galois
Hasse_norm_theorem
Fantasy comedy TV series (2019–2026)
narrates the series. All episodes of the first series were released on Amazon Prime Video on 31 May 2019, and aired weekly on BBC Two in the UK between 15 January
Good_Omens_(TV_series)
Unique numeric book identifier since 1970
registrant element, the publication element, and a checksum character or check digit. A 13-digit ISBN can be separated into its parts (prefix element, registration
ISBN
Montenegrin politician (born 1965)
partners the People's Party and Democratic Serb Party, introducing the prime element of European Social Democracy in SNP's political ideology. He rendered
Srđan_Milić
Complex number whose real and imaginary parts are both integers
every product of an element of I by an element of R belong to I. An ideal is principal if it consists of all multiples of a single element g, that is, it has
Gaussian_integer
British political leadership election
leadership election rules that had been introduced in 1993, which included an element of one member, one vote. The poll for leader was held simultaneously with
1994 Labour Party leadership election
1994_Labour_Party_leadership_election
Minimal element in the set of prime ideals ordered by inclusion
decreasing chain of prime ideals is prime. Therefore, the set of prime ideals containing I has a minimal element, which is a minimal prime over I. Emmy Noether
Minimal_prime_ideal
In mathematics, a square-free element is an element r of a unique factorization domain R that is not divisible by a non-trivial square. This means that
Square-free_element
International standards for dates and times
standard does not assign specific meaning to any element of the dates/times represented: the meaning of any element depends on the context of its use. Dates and
ISO_8601
American action thriller television series
Hopper. The Terminal List: Dark Wolf premiered on August 27, 2025, on Amazon Prime Video. Taylor Kitsch as Chief Special Warfare Operator Benjamin "Ben" Edwards
The_Terminal_List:_Dark_Wolf
Algebraic structure with addition, multiplication, and division
existence of an additive inverse −a for each element a and of a multiplicative inverse b−1 for each nonzero element b. This allows the definition of the so-called
Field_(mathematics)
PRIME ELEMENT
PRIME ELEMENT
Boy/Male
Arabic, Muslim
Prime Minister
Boy/Male
Welsh American
Son of Rhys.
Girl/Female
Australian, French, German, Italian, Latin, Swedish
First-born
Boy/Male
Australian, British, Christian, English, Welsh
Son of Rhys; Ardent; Son of the Ardent; Prize
Male
Italian
Italian and Spanish form of Latin Primus, PRIMO means "first."
Boy/Male
Arabic, Muslim, Pashtun
Prime Chief
Girl/Female
Latin
Firstborn.
Female
English
English name derived from Latin prima, PRIMULA means "first, prime."
Surname or Lastname
English
English : unexplained.Serbian : unexplained.
Surname or Lastname
English
English : from the Old Norse personal name GrÃmr, which remained popular as a personal name in the form Grim in Anglo-Scandinavian areas well into the 12th century. It was a byname of Woden with the meaning ‘masked person’ or ‘shape-changer’, and may have been bestowed on male children in an attempt to secure the protection of the god. The Continental Germanic cognate grÄ«m was also used as a first element in compound names. Compare Grimaud and Gribble, with the original sense ‘mask’, ‘helmet’. Some examples of the surname may derive from short forms of such names.
Boy/Male
Muslim
Prime minister
Girl/Female
Hindu
Love, Affection
Boy/Male
Indian, Tamil
Important; Prime
Surname or Lastname
Welsh
Welsh : Anglicized form of Welsh ap Rhys ‘son of Rhys’ (see Reece). This is one of the commonest of Welsh surnames. It has also been established in Ireland since the 14th century, where it is sometimes a variant of Bryson.English : the name is also found very early in parts of England far removed from Welsh influence (e.g. Richard Prys, Essex 1320), and in such cases presumably derives from Middle English, Old French pris ‘price’, ‘prize’, perhaps as a metonymic occupational name for a fixer of prices.Americanized spelling of Jewish Preuss or Preis.
Surname or Lastname
German
German : of uncertain origin; possibly from the Latin personal name Primus (‘the first’), borne by several saints; or one composed with a Germanic word meaning ‘to prick or stab’; or from a personal name of Slavic origin Primm, from prēmu ‘right’.French : from a personal name (from Latin Primus).French : nickname from Old French prim ‘first’, possibly given to the eldest child in a family, or alternatively a nickname from Old French and Occitan prim ‘shrewd’, ‘clever’, ‘artful’, ‘sly’.Dutch : variant of Priem.English : variant of Prime.Some of the Prim families in VT descend from a Simon Laval dit Printemps, who was known in English-speaking areas as Seymour Prim.
Surname or Lastname
English
English : from a Middle English personal name or nickname. The personal name existed in Old English, and is probably derived from Old English prim ‘early morning’ (from Latin primus ‘first’, used as the name of one of the canonical hours). The surname may be derived from this word as a Middle English nickname in the sense ‘fine’, ‘excellent’.French : feminine form of Prim 3.Dutch : variant of Priem.Probably an Americanized spelling of German Preim, a topographic name (of Slavic origin), perhaps from a river near Hannover; or of Preime, a variant of Primus.
Male
English
English surname transferred to forename use, derived from the Middle English element pris, PRICE means "price" or "prize."Â
Boy/Male
Hindu, Indian
Chief; Prime
Boy/Male
Indian
Prime
Boy/Male
Indian, Italian, Latin
First Born
PRIME ELEMENT
PRIME ELEMENT
Boy/Male
Arabic
Freedom from Sorrow
Boy/Male
Hindu, Indian
Yadav
Girl/Female
Greek American Hebrew Italian Spanish
From the Hebrew Elisheba, meaning either oath of God, or God is satisfaction. Famous bearer: Old...
Boy/Male
Muslim
Worshipers
Boy/Male
Hindu, Indian, Kannada, Telugu
Fixed
Girl/Female
Tamil
Male
Welsh
Variant spelling of Welsh Cadwalader, CADWALLADER means "battle leader."
Girl/Female
Tamil
Satyarupa | ஸதà¯à®¯à®¾à®°à¯‚பா
Dedicated to truth, Personified
Boy/Male
English American French Latin
Lathe worker.
Girl/Female
Indian, Tamil
Warrior Women of History; Shiva's Lover
PRIME ELEMENT
PRIME ELEMENT
PRIME ELEMENT
PRIME ELEMENT
PRIME ELEMENT
n.
See Price, and 1st Prize.
n.
Highest pitch; elevation reached; loftiness; prime; glory; as, to be in the pride of one's life.
a.
First in rank, degree, dignity, authority, or importance; as, prime minister.
a.
Being in its prime.
v. t.
To set a price on; to value. See Prize.
n.
One who, or that which, primes
#
Donne (#) (pl. ) of Prima donna
n. & v.
See Prize, n., 5. Also Prize, v. t.
n. & v.
Reward; recompense; as, the price of industry.
a.
To mark with a prime mark.
a.
To prepare; to make ready; to instruct beforehand; to post; to coach; as, to prime a witness; the boys are primed for mischief.
a.
Marked or distinguished by a mark (') called a prime mark.
imp. & p. p.
of Prime
a.
A prime number. See under Prime, a.
v. t.
To ask the price of; as, to price eggs.
n.
That which occasion crime.
a.
To lay the first color, coating, or preparation upon (a surface), as in painting; as, to prime a canvas, a wall.
a.
First in excellence; of highest quality; as, prime wheat; a prime quality of cloth.
v. t.
To pay the price of.
a.
Formal; precise; affectedly neat or nice; as, prim regularity; a prim person.