By Joseph J. Rotman

This book's organizing precept is the interaction among teams and earrings, the place “rings” contains the tips of modules. It includes uncomplicated definitions, whole and transparent theorems (the first with short sketches of proofs), and provides consciousness to the themes of algebraic geometry, desktops, homology, and representations. greater than simply a succession of definition-theorem-proofs, this article placed effects and concepts in context in order that scholars can get pleasure from why a undeniable subject is being studied, and the place definitions originate. bankruptcy issues comprise teams; commutative jewelry; modules; primary excellent domain names; algebras; cohomology and representations; and homological algebra. for people attracted to a self-study advisor to studying complex algebra and its similar subject matters.

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Conversely, let {Ai : i ∈ I } be a partition of X . If x, y ∈ X , define x ≡ y if there is i ∈ I with both x ∈ Ai and y ∈ Ai . It is plain that ≡ is reflexive and symmetric. To see that ≡ is transitive, assume that x ≡ y and y ≡ z; that is, there are i, j ∈ I with x, y ∈ Ai and y, z ∈ A j . Since y ∈ Ai ∩ A j , pairwise disjointness gives Ai = A j , so that i = j and x, z ∈ Ai ; that is, x ≡ z. We have shown that ≡ is an equivalence relation. It remains to show that the equivalence classes are the Ai ’s.

Remark. 59 on page 36 shows that if both f and g are injective, then so is their composite f ◦ g. Similarly, f ◦ g is a surjection if both f and g are surjections. It follows that the composite of two bijections is itself a bijection. Notation. 54 on page 36 says that a function cannot have two inverses). 48. Here is an example of two functions f and g one of whose composites g ◦ f is the identity while the other composite f ◦ g is not the identity; thus, f and g are not inverse functions. If N = {n ∈ Z : n ≥ 0}, define f , g : N → N as follows: f (n) = n + 1; g(n) = 0 if n = 0 n − 1 if n ≥ 1.

An equivalence relation on a set X yields a family of subsets of X . Definition. Let ≡ be an equivalence relation on a set X . If a ∈ X , the equivalence class of a, denoted by [a], is defined by [a] = {x ∈ X : x ≡ a} ⊆ X. For example, under congruence mod m, the equivalence class [a] of an integer a is called its congruence class. The next lemma says that we can replace equivalence by honest equality at the cost of replacing elements by their equivalence classes. Sec. 53. [x] = [y]. 35 If ≡ is an equivalence relation on a set X , then x ≡ y if and only if Proof.