By Joseph J. Rotman

E-book DescriptionThis book's organizing precept is the interaction among teams and earrings, the place "rings" comprises the tips of modules. It includes simple definitions, whole and transparent theorems (the first with short sketches of proofs), and provides realization to the themes of algebraic geometry, pcs, homology, and representations. greater than in simple terms a succession of definition-theorem-proofs, this article placed effects and concepts in context in order that scholars can take pleasure in why a undeniable subject is being studied, and the place definitions originate. bankruptcy themes comprise teams; commutative jewelry; modules; vital excellent domain names; algebras; cohomology and representations; and homological algebra. for people drawn to a self-study advisor to studying complicated algebra and its similar topics.Book information includes easy definitions, entire and transparent theorems, and provides cognizance to the themes of algebraic geometry, desktops, homology, and representations. for people attracted to a self-study consultant to studying complicated algebra and its similar themes.

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**Example text**

To prove pairwise disjointness, assume that a ∈ [x] ∩ [y], so that a ≡ x and a ≡ y. By symmetry, x ≡ a, and so transitivity gives x ≡ y. Therefore, [x] = [y], by the lemma, and the equivalence classes form a partition of X . 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 .

Definition. A function f : X → Y is a surjection (or is onto) if im f = Y. Thus, f is surjective if, for each y ∈ Y , there is some x ∈ X (probably depending on y) with y = f (x). The following definition gives another important property a function may have. Definition. A function f : X → Y is an injection (or is one-to-one) if, whenever a and a are distinct elements of X , then f (a) = f (a ). Equivalently (the contrapositive states that) f is injective if, for every pair a, a ∈ X , we have f (a) = f (a ) implies a = a .

Iv) If s ∈ S, then f (s) ∈ f (S), and so s ∈ f −1 f (s) ⊆ f −1 f (S). To see that there may be strict inclusion, let f : R → C be given by x → e2πi x . If S = {0}, then f (S) = {1} and f −1 f ({1}) = Z. 68 on page 37, we will see that if f : X → Y , then inverse image behaves better on subsets than does forward image; for example, f −1 (S ∩ T ) = f −1 (S) ∩ f −1 (T ), where S, T ⊆ Y , but for A, B ⊆ X , it is possible that f (A ∩ B) = f (A) ∩ f (B). We will need cartesian products of more than two sets.