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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:
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Definition 10.3 Direct Sum
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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:
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Theorem 10.7 Direct Sums of n Subgroups
Theorem 10.8 Direct Sums and Isomorphisms
Proof:
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Rings
Definition 10.9a Definition of a Ring
Note:
Definition 10.9b Alternative Definition of a Ring
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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
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Definition 10.13 Ring with Unity, Commutative Ring
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Theorem 10.14 Uniquness of the Unity
Definition 10.15 Multiplicative Inverse
Theorem 10.16 Uniquness of the Multiplicative Inverse
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Theorem 10.17 Zero Product
Theorem 10.18 Zero Divisor
Theorem 10.19 Additive Inverses and Products
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Theorem 10.20 Generalized Associative Laws
Theorem 10.21 Generalized Distributive Laws