Lecture32-Mar10

Lecture 32 Overview

• Date: March 10

Group Quotient Construction

• Completed discussion on the quotient construction for commutative rings.

• Began exploration of groups with the following Proposition:

  • Let m be any equivalence relation on group G (denoted as 6).

  • Define the operation: [3][4] = [gh]; this creates a group structure on G/∼.

• Important Note:

  • guh when gh is in [16], this aligns with the construction of the group quotient G/[1].

  • Only way to define a group structure on equivalence classes is through the group quotient.

Example of Equivalence Relations

• Different equivalence relations exist for a given group G.

  • Let s be a subgroup of G. Define grah iff h is in s, giving a possible group structure.

  • This structure on G/s exists only if s is a normal subgroup.

    • If s is normal, it coincides with quotient construction; otherwise, it does not.

Commutative Rings and Ideals

• Let R be a commutative ring.

• Theorem I regarding subgroups:

  • A subset S of R is an equivalence relation iff S is a subgroup using addition.

  • Operations on R/S:

    • + [y] = [x+y]

    • * [y] = [x*y]

  • S must be an ideal for these operations to define a ring structure on R/S.

Understanding 'Ideal'

• In ring theory, "ideal" has similar importance as a normal subgroup does in group theory.

• Historical context: Mathematicians sought to generalize number theory (e.g., unique factorization in integers) into the realm of rings.

• Different examples of rings:

  • Complex integers: R = a + bi

  • More complex numbers: K = a + b√2 + c√3, where a, b, c are integers.

• Unique factorization doesn’t hold for all rings, prompting the concept of ideals in number theory.

Geometry and Modern Algebra

• In geometry, curves or surfaces correspond to quotient rings.

  • Example: Circle identification through the equation x² + y² = 1.

• Points on the circle relate to ideals within the ring R.

• This reflects the modern treatment of number theory through the lens of rings, ideals, and quotients.

• Denoting R/S as the quotient construction provides clarity and understanding of structures across various mathematical fields.

Proofs and Assignments

• The proof of Theorem I mirrors those seen in vector spaces and group theory.

• Assignment reference: Examine what "ideal" means when K' is not commutative.

Example of Ideals

• Example given:

  • R = Z (integers)

  • S = nZ is an ideal.

  • The resultant quotient R/S corresponds to Z/nZ.

• Highlights the correct representation of R/S not only as a ring but sometimes as a field, emphasizing equivalence classes: [r] = r + S.

• Reminders for calculations: Select good representatives, and remember to "mod by S" to simplify calculations.