Lecture 10 – Direct Sums and Rings

Direct Sums

Definition 10.1 Sums of Subgroups

Theorem 10.2 Sums of Subgroups

Proof:

Note:

Definition 10.3 Direct Sum

Theorem 10.4 Order of a Direct Sum

Theorem 10.5 Equivalent Condition for a Direct Sum

Proof:

Theorem 10.6 Sums of Two Subgroups

Proof:

Theorem 10.7 Direct Sums of n Subgroups

Theorem 10.8 Direct Sums and Isomorphisms

Proof:

Rings

Definition 10.9a Definition of a Ring

Note:

Definition 10.9b Alternative Definition of a Ring

Example 2: The set E of all even integers is a ring with respect to the usual addition and multiplication in Z.

Definition 10.10 Subring

Theorem 10.11 Equivalent Set of Conditions for a Subring

Theorem 10.12 Characterization of a Subring

Definition 10.13 Ring with Unity, Commutative Ring

Theorem 10.14 Uniquness of the Unity

Definition 10.15 Multiplicative Inverse

Theorem 10.16 Uniquness of the Multiplicative Inverse

Theorem 10.17 Zero Product

Theorem 10.18 Zero Divisor

Theorem 10.19 Additive Inverses and Products

Theorem 10.20 Generalized Associative Laws

Theorem 10.21 Generalized Distributive Laws