ASL STEM

Product Ring Sign Video

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Definition

Let R and R' be rings. The product set RxR' is a ring called the product ring, with component-wise addition and multiplication: (x,x')+(y,y')=(x+y,x'+y') and (x,x')(y,y')=(xy,x'y'). The additive and multiplicative identities in RxR' are (0,0) and (1,1), respectively.

Source: Algebra, second edition by Michael Artin

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> Mathematics

> Abstract Algebra

> Product Ring

+ Characteristic

+ Symmetric Group

+ Cancellation Law *

+ Trivial Subgroup *

+ General Linear Group *

+ Induced Law *

+ Lagrange's Theorem *

+ Special Linear Group *

+ Proper Subgroup *

+ Law Of Composition *

+ Product Set *

+ Transposition *

+ Abelian Group

+ Infinite Cyclic *

+ Quaternion Group *

+ Klein Four Group *

+ Cyclic Subgroup

+ Skew Symmetric *

+ Trivial Homomorphism *

+ Inclusion Map *

+ Sign Homomorphism *

+ Alternating Group *

+ Normal Subgroup

+ Isomorphism Class *

+ Automorphism *

+ Right Coset *

+ Subgroup Index *

+ Correspondence Theorem *

+ Counting Formula *

+ Conjugate Subgroup *

+ Group Operation *

+ Product Group *

+ Quotient Group *

+ Canonical Map *

+ Permutation Representation *

+ Cayley's Theorem *

+ Conjugacy Class *

+ Centralizer *

+ Class Equation *

+ Self Adjoint *

+ Subdiagonal *

+ Superdiagonal *

+ Commutator Subgroup *

+ Fixed Vector *

+ First Isomorphism Theorem *

+ Proper Subspace *

+ Prime Field *

+ Characteristic Zero *

+ Primitive Root *

+ Jordan Block *

+ Unitary Group *

+ Hypervector *

+ Linear Relation *

+ Basechange Matrix *

+ Dimension Formula *

+ Jordan Canonical Form *

+ Jordan Decomposition Theorem *

+ Kronecker Delta *

+ Orthogonal Group *

+ Special Orthogonal Group *

+ Orthogonal Operator *

+ Axis Of Rotation *

+ Rotation Matrix *

+ Rotation Group *

+ Cayley Hamilton Theorem *

+ Matrix Exponential *

+ Unitary Matrix *

+ First Sylow Theorem *

+ Linear Group *

+ Dihedral Group *

+ Translation Group *

+ Discrete Subgroup *

+ Representation *

+ Point Group *

+ Tetrahedral Group *

+ Octahedral Group *

+ Icosahedral Group *

+ Truncated Polyhedron *

+ Sylow P Subgroup *

+ Second Sylow Theorem *

+ Third Sylow Theorem *

+ Bilinear Form *

+ Positive Definite *

+ Negative Semidefinite *

+ Negative Definite *

+ Hermitian Form *

+ Standard Hermitian Form *

+ Hermitian Symmetry *

+ Positive Semidefinite *

+ Nondegenerate *

+ Orthogonal Sum *

+ Orthogonal Projection *

+ Hermitian Space *

+ Adjoint Operator *

+ Symmetric Operator *

+ Degenerate Conic *

+ Nondegenerate Conic *

+ Orthogonal Vector *

+ Quadric Surface *

+ Symplectic Group *

+ Lorentz Group *

+ Lorentz Transformations *

+ Special Unitary Group *

+ Quaternion Algebra *

+ Skew Hermitian *

+ Spin Homomorphism *

+ Orthogonal Representation *

+ Maschke's Theorem *

+ One Parameter Group *

+ Gaussian Integers *

+ Jacobi Identity *

+ Lie Algebra *

+ Irreducible Character *

+ Complex Algebraic Group *

+ Exceptional Groups *

+ Sporadic Group *

+ Matrix Representation *

+ Trivial Representation *

+ Group Character *

+ Mathieu Group *

+ Monster Group *

+ Regular Representation *

+ Proper Divisor *

+ Compact Group *

+ Noncommutative Ring *

+ Leading Coefficient *

+ Total Degree *

+ Homogeneous Polynomial *

+ Ring Homomorphism *

+ Principal Ideal *

+ Proper Ideal *

+ Ring Extension *

- Product Ring *

+ Zero Divisor *

+ Fraction Field *

+ Maximal Ideal *

+ Integral Domain

+ Hilbert's Nullstellensatz *

+ Common Zeros *

+ Algebraic Variety *

+ Bezout Bound *

+ Riemann Surface *

+ Plane Algebraic Curve *

+ Cyclotomic Polynomial *

+ Branched Covering *

+ Unbranched Covering *

+ Branch Points *

+ Size Function *

+ Euclidean Domain *

+ Principal Ideal Domain *

+ Irreducible Factorization *

+ Unique Factorization Domain *

+ Gauss's Lemma *

+ Sieve Of Eratosthenes *

+ Eisenstein Criterion *

+ Gaussian Prime *

+ Algebraic Integer *

+ Quadratic Number Field *

+ Ring Of Integers *

+ Half Integer *

+ Ideal Class *

+ Class Number *

+ Product Ideal *

+ Prime Ideal *

+ Class Group *

+ Free Module *

+ Invertible Integer Matrix *

+ Elementary Integer Matrix *

+ Complete Set *

+ Irreducible Representation *

+ Quotient Ring *

+ Ascending Chain Condition *

+ Noetherian Ring *

+ Hilbert Basis Theorem *

+ Rational Canonical Form *

+ Extension Field *

+ Algebraic Number Field *

+ Finite Field *

+ Function Field *

+ Finite Extension *

+ Primitive Element *

+ Algebraically Closed *

+ Symmetric Polynomial *

+ Sylvester's Inertia Law *

+ Sylvester's Signature *

+ Projective Group *

+ Elementary Symmetric Functions *

+ Symmetric Functions Theorem *

+ Discriminant *

+ Tschirnhausen Transformation *

+ Splitting Field *

+ Splitting Theorem *

+ Galois Group *

+ Galois Extension Field *

+ Fixed Field *

+ Fixed Field Theorem *

+ Luroth's Theorem *

+ Intermediate Field *

+ Crystallographic Point Group *

+ Free Semigroup *

+ Resolvent Cubic *

+ Cyclotomic Field *

+ Kronecker Weber Theorem *

+ Kummer Extension *

+ Linear Operator *

+ Invertible Operator *

+ Singular Operator *

+ Shift Operator *

+ Commutative Ring

+ Generalized Eigenvector *

+ Schur's Lemma *

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