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Basic Algebra I.5 (Operations on Ideals)
Given ideals I and J of ring R, we can perform the following operations to obtain new ideals: is an ideal of R; is an ideal of R; is an ideal of R. Thus, IJ is the set of all … Continue reading
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Tagged abstract algebra, algebra, basic course, notes, ring theory, undergraduate
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Basic Algebra I.4 (Ideals and Ring Quotients)
Our first naïve attempt is to take a ring quotient R/S for a subring S of R. First, (S, +) is a subgroup of (R, +) so we can denote every coset R/S by x+S for some x in R. … Continue reading
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Tagged abstract algebra, algebra, basic course, notes, ring theory, undergraduate
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Basic Algebra I.3 (Subrings and Ring Products)
The concept of subrings follow naturally. Let R be a ring. A subring is a subset S of R such that: (S, +) is a subgroup of (R, +); S contains 1; for any a, b in S, ab is … Continue reading
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Tagged abstract algebra, algebra, basic course, notes, ring theory, undergraduate
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Basic Algebra I.2 (Examples + Basic Properties of Rings)
Examples The set Z of integers forms a ring under addition and multiplication, but the subset 2Z of even integers forms a rng. The set Z/nZ of integers modulo n forms a ring under addition and multiplication mod n. We have rings Q, R and C, which are … Continue reading
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Tagged abstract algebra, algebra, basic course, notes, ring theory, undergraduate
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Basic Algebra I.1 (Enter the Rings)
In this series, we will cover some common algebraic structures – other than groups, which had been amply covered in the Group Theory series. However, due to the considerable depth of many of the topics, we can only provide a brief … Continue reading
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Tagged abstract algebra, algebra, basic course, notes, ring theory, undergraduate
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Group Theory ∞ (Epilogue)
That concludes the end of the series of notes on Group Theory. Has it been successful? I don’t know, but I’m reasonably pleased with the way the notes turn out, except the fact that there’s a huge disparity between the … Continue reading
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Tagged abstract algebra, algebra, basic course, epilogue, group theory, notes, undergraduate
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Group Theory XII.5 (More Category Theory)
Since a category is really a bunch of abstract objects and arrows between them, we can reverse them by duality. Let C be a category. The opposite category Cop is the category such that: Ob(Cop) = Ob(C); for any objects … Continue reading
Posted in Group Theory Notes
Tagged abstract algebra, algebra, basic course, group theory, notes, undergraduate
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