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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