Quotient groups and quotient rings

Se basa en

Requisitos previos obligatorios, hasta el fondo: 18

Aprenderás a

• Construct G/N and R/I and prove that the operations are well defined exactly when N is normal, respectively when the additive subgroup I is an ideal.
• Prove the first isomorphism theorem and use it to identify quotients such as ℤ/nℤ, ℝ[x]/(x² + 1) ≅ ℂ and GLₙ(k)/SLₙ(k) ≅ kˣ.
• Show that R/I is a field if and only if I is maximal, and an integral domain if and only if I is prime.

Definición

• For a normal subgroup N of G, G/N is the set of cosets gN with (gN)(hN) = ghN.
• For an ideal I of a ring R, R/I is the set of cosets a + I with the induced addition and multiplication.
• The quotient map G → G/N sends g to gN.

Teorema

• First isomorphism theorem: for a homomorphism φ : G → H, G/ker φ ≅ im φ, and likewise for rings.
• ℤ/nℤ, the quotient of ℤ by the ideal nℤ, is the ring of congruence classes modulo n.
• For a commutative ring R with identity, R/I is a field if and only if I is a maximal ideal.
• Universal property: every homomorphism G → H whose kernel contains N factors uniquely through the quotient map G → G/N.

Errores frecuentes

• Thinking G/N is a subgroup of G.
• Defining operations on cosets without checking that they are independent of the chosen representatives.
• Assuming every non-zero quotient ℤ/nℤ is a field.

Conduce a

Descubrimientos opcionales

• Limits and colimits The quotient G → G/N is a coequaliser in the category of groups (of the inclusion N → G and the trivial homomorphism); quotients in general are colimits.Sendero del Asombro →
• ← Relations, equivalence relations and partitions
• ← Congruences

Itinerarios de aprendizaje

• Undergraduate mathematics major
• Graduate preparation

Lecturas complementarias

• David S. Dummit and Richard M. Foote, Abstract Algebra
• Michael Artin, Algebra (1991)