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Algebraic Structures
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Kernel of Group Homomorphism
The set of elements in the domain of the homomorphism that are mapped to the identity element of the codomain.
Cyclic Group
A group that can be generated by a single element, meaning that every element in the group can be written as some power of this generator.
Characteristic of a Ring
The smallest positive number of times the identity under addition must be added to itself to get zero; if there is no such number, the ring has characteristic zero.
Subring
A subset of a ring that is itself a ring with the same addition and multiplication as the larger ring.
Ring
Set with two binary operations (addition and multiplication) satisfying certain conditions, including: a group under addition, distributive property of multiplication over addition, and multiplication is associative.
Group Isomorphism
A bijective group homomorphism; a structure-preserving one-to-one correspondence between two groups.
Field Extension
A larger field containing a smaller field as a subfield; algebraic structures important in the study of field theory and Galois theory.
Simple Group
A nontrivial group whose only normal subgroups are the trivial group and the group itself.
Normal Subgroup
A subgroup of a group such that for all elements in and in , the element is also in .
Field
Set on which two operations, addition and multiplication, are defined and behave as they do with rational and real numbers, including the existence of additive and multiplicative inverses (except for the additive identity).
Galois Group
The group of field automorphisms (bijection from a field to itself that preserves the algebraic operations) that leave a certain smaller field fixed, important in understanding the solvability of polynomial equations.
Algebra over a Field
A ring that also has the structure of a vector space over a field, together with a bilinear product.
Free Group
A group formed by the set of all possible words that can be built from members of a given set, called generators, subject to the only relations being that a generator composed with its inverse gives the identity element.
Group Homomorphism
A function between two groups that preserves the group operation: for all in the group.
Quotient Group
The group consisting of the cosets of a normal subgroup, with the operation defined by the product of cosets.
Coset
The set of all products of a particular element from the group with all of the elements of a particular subgroup.
Vector Space
A collection of vectors that can be scaled and added together to produce another vector in the same space, satisfying axioms related to vector addition and scalar multiplication.
Direct Product of Groups
The group constructed from two groups, whose elements are ordered pairs from the two groups, with the group operation defined componentwise.
Ideal
A subset of a ring that is itself a subring and is closed under multiplication by elements from the ring.
Ring Homomorphism
A function between two rings that respects both the addition and multiplication operations of the rings.
Abelian Group
A group where the group operation is commutative, meaning that for any two elements and in the group, .
Group
Set equipped with a single operation that combines any two elements to form a third element, satisfying four conditions: closure, associativity, identity, and inversibility.
Module
A generalization of a vector space, except the scalars come from a ring instead of a field.
Left Ideal
A nonempty subset of a ring that is both an additive subgroup and closed under multiplication by any element of the ring on the left.
Commutator
Given two elements and of a group, the commutator is the element .
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