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93 relations: Abstract algebra, Additive group, Annihilator (ring theory), Artinian ideal, Bimodule, Boolean prime ideal theorem, Bounded operator, Chinese remainder theorem, Closure (mathematics), Compact operator, Complete lattice, Congruence relation, Continuous function, Coset, Cyclic group, David Hilbert, Dedekind domain, Distributive lattice, Division ring, Emmy Noether, Empty set, Empty sum, Ernst Kummer, Field (mathematics), Field of fractions, Finitely generated module, Fractional ideal, Fundamental theorem of arithmetic, Generating set of a module, Group theory, Homomorphism, Ideal (order theory), Ideal norm, Ideal number, Ideal quotient, Ideal theory, Idealizer, Integer, Integral domain, Intersection (set theory), Irreducible ideal, Isomorphism theorems, Kernel (algebra), Krull's theorem, Lattice (order), Macaulay computer algebra system, Matrix (mathematics), Maximal ideal, Michiel Hazewinkel, Minimal ideal, ..., Modular arithmetic, Modular lattice, Module (mathematics), Natural number, Nil ideal, Noncommutative ring, Normal subgroup, Number theory, Opposite ring, Order theory, Parity (mathematics), Partially ordered set, Pointwise product, Polynomial, Primary ideal, Prime ideal, Prime number, Prime ring, Primitive ideal, Primitive ring, Principal ideal, Principal ideal domain, Pseudo-ring, Quotient group, Quotient ring, Radical of an ideal, Reduced ring, Regular ideal, Richard Dedekind, Ring (mathematics), Ring homomorphism, Ring theory, Semiprime ring, Semiring, Simple module, Simple ring, Springer Science+Business Media, Subgroup, Subset, Union (set theory), Unit (ring theory), Vorlesungen über Zahlentheorie, Zorn's lemma. Expand index (43 more) »
Abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.
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Additive group
An additive group is a group of which the group operation is to be thought of as addition in some sense.
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Annihilator (ring theory)
In mathematics, specifically module theory, the annihilator of a set is a concept generalizing torsion and orthogonality.
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Artinian ideal
In abstract algebra, an Artinian ideal, named after Emil Artin, is encountered in ring theory, in particular, with polynomial rings.
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Bimodule
In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible.
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Boolean prime ideal theorem
In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given algebra.
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Bounded operator
In functional analysis, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded above by the same number, over all non-zero vectors v in X. In other words, there exists some M\ge 0 such that for all v in X The smallest such M is called the operator norm \|L\|_ \, of L. A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless L(v).
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Chinese remainder theorem
The Chinese remainder theorem is a theorem of number theory, which states that if one knows the remainders of the Euclidean division of an integer by several integers, then one can determine uniquely the remainder of the division of by the product of these integers, under the condition that the divisors are pairwise coprime.
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Closure (mathematics)
A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation.
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Compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset (has compact closure) of Y. Such an operator is necessarily a bounded operator, and so continuous.
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Complete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet).
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Congruence relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure.
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Continuous function
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
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Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then Only when H is normal will the set of right cosets and the set of left cosets of H coincide, which is one definition of normality of a subgroup.
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Cyclic group
In algebra, a cyclic group or monogenous group is a group that is generated by a single element.
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David Hilbert
David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician.
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Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals.
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Distributive lattice
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other.
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Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible.
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Emmy Noether
Amalie Emmy NoetherEmmy is the Rufname, the second of two official given names, intended for daily use.
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Empty set
In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.
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Empty sum
In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero.
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Ernst Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician.
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Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
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Field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded.
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Finitely generated module
In mathematics, a finitely generated module is a module that has a finite generating set.
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Fractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains.
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Fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.
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Generating set of a module
In algebra, a generating set G of a module M over a ring R is a subset of M such that the smallest submodule of M containing G is M itself (the smallest submodule containing a subset is the intersection of all submodules containing the set).
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Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.
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Homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).
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Ideal (order theory)
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset).
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Ideal norm
In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension.
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Ideal number
In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings.
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Ideal quotient
In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient (I: J) is the set Then (I: J) is itself an ideal in R. The ideal quotient is viewed as a quotient because IJ \subseteq K if and only if I \subseteq K: J. The ideal quotient is useful for calculating primary decompositions.
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Ideal theory
In mathematics, ideal theory is the theory of ideals in commutative rings; and is the precursor name for the contemporary subject of commutative algebra.
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Idealizer
In abstract algebra, the idealizer of a subsemigroup T of a semigroup S is the largest subsemigroup of S in which T is an ideal.
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Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
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Integral domain
In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.
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Intersection (set theory)
In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.
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Irreducible ideal
In mathematics, a proper ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two strictly larger ideals.
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Isomorphism theorems
In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects.
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Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective.
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Krull's theorem
In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ring has at least one maximal ideal.
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Lattice (order)
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.
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Macaulay computer algebra system
Macaulay is a computer algebra system for doing polynomial computations, particularly Gröbner basis calculations.
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Matrix (mathematics)
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
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Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals.
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Michiel Hazewinkel
Michiel Hazewinkel (born 22 June 1943) is a Dutch mathematician, and Emeritus Professor of Mathematics at the Centre for Mathematics and Computer and the University of Amsterdam, particularly known for his 1978 book Formal groups and applications and as editor of the Encyclopedia of Mathematics.
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Minimal ideal
In the branch of abstract algebra known as ring theory, a minimal right ideal of a ring R is a nonzero right ideal which contains no other nonzero right ideal.
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Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).
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Modular lattice
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition:;Modular law: x ≤ b implies x ∨ (a ∧ b).
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Module (mathematics)
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.
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Natural number
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").
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Nil ideal
In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent.
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Noncommutative ring
In mathematics, more specifically abstract algebra and ring theory, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exists a and b in R with a·b ≠ b·a.
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Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.
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Number theory
Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.
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Opposite ring
In algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order.
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Order theory
Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations.
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Parity (mathematics)
In mathematics, parity is the property of an integer's inclusion in one of two categories: even or odd.
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Partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.
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Pointwise product
The pointwise product of two functions is another function, obtained by multiplying the image of the two functions at each value in the domain.
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Polynomial
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
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Primary ideal
In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n>0.
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Prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers.
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Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
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Prime ring
In abstract algebra, a nonzero ring R is a prime ring if for any two elements a and b of R, arb.
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Primitive ideal
In mathematics, a left primitive ideal in ring theory is the annihilator of a simple left module.
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Primitive ring
In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module.
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Principal ideal
In the mathematical field of ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x of P, which is to say the set of all elements less than or equal to x in P. The remainder of this article addresses the ring-theoretic concept.
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Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element.
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Pseudo-ring
In mathematics, and more specifically in abstract algebra, a pseudo-ring is one of the following variants of a ring.
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Quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.
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Quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra.
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Radical of an ideal
In commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x is in the radical if some power of x is in I. A radical ideal (or semiprime ideal) is an ideal that is its own radical (this can be phrased as being a fixed point of an operation on ideals called 'radicalization').
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Reduced ring
In ring theory, a ring R is called a reduced ring if it has no non-zero nilpotent elements.
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Regular ideal
In mathematics, especially ring theory, a regular ideal can refer to multiple concepts.
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Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers.
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Ring (mathematics)
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
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Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.
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Ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers.
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Semiprime ring
In ring theory, a branch of mathematics, semiprime ideals and semiprime rings are generalizations of prime ideals and prime rings.
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Semiring
In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.
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Simple module
In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that have no non-zero proper submodules.
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Simple ring
In abstract algebra, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Subgroup
In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.
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Subset
In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.
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Union (set theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.
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Unit (ring theory)
In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.
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Vorlesungen über Zahlentheorie
Vorlesungen über Zahlentheorie (German for Lectures on Number Theory) is the name of several different textbooks of number theory.
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Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max Zorn and Kazimierz Kuratowski, is a proposition of set theory that states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element.
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Algebraic ideal, Comaximal, Finitely generated ideal, Ideal (algebra), Ideal (ring), Ideal (rings), Ideal Product, Ideal of a ring, Ideal ring theory, Left ideal, One-sided ideal, Product of ideals, Proper ideal, Right ideal, Ring ideal, Sum of ideals, Two sided ideal, Two-sided ideal, Unit ideal.
References
[1] https://en.wikipedia.org/wiki/Ideal_(ring_theory)
