By Joseph J. Rotman

Publication DescriptionThis book's organizing precept is the interaction among teams and earrings, the place "rings" contains the tips of modules. It includes uncomplicated definitions, entire and transparent theorems (the first with short sketches of proofs), and offers realization to the subjects of algebraic geometry, pcs, homology, and representations. greater than only a succession of definition-theorem-proofs, this article placed effects and concepts in context in order that scholars can have fun with why a definite subject is being studied, and the place definitions originate. bankruptcy subject matters contain teams; commutative jewelry; modules; crucial excellent domain names; algebras; cohomology and representations; and homological algebra. for people drawn to a self-study advisor to studying complicated algebra and its comparable topics.Book details includes simple definitions, entire and transparent theorems, and offers realization to the subjects of algebraic geometry, pcs, homology, and representations. for people drawn to a self-study consultant to studying complex algebra and its comparable subject matters.

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**Example text**

The addition theorem and De Moivre’s theorem can be restated in complex exponential form. The first becomes ei x ei y = ei(x+y) ; the second becomes (ei x )n = einx . Definition. ζ n = 1. If n ≥ 1 is an integer, then an nth root of unity is a complex number ζ with The geometric interpretation of complex multiplication is particularly interesting when z and w lie on the unit circle, so that |z| = 1 = |w|. Given a positive integer n, let θ = 2π/n and let ζ = eiθ . The polar coordinates of ζ are (1, θ), the polar coordinates of ζ 2 are (1, 2θ ), the polar coordinates of ζ 3 are (1, 3θ ),.

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 . The reader should note that being injective is the converse of being well-defined: f is well-defined if a = a implies f (a) = f (a ); f is injective if f (a) = f (a ) implies a=a. There are other names for these functions. Surjections are often called epimorphisms and injections are often called monomorphisms.

Definition. A function f : X → Y is a bijection (or a one-to-one correspondence) if it is both an injection and a surjection. Sec. 3 Some Set Theory 31 Definition. A function f : X → Y has an inverse if there is a function g : Y → X with both composites g ◦ f and f ◦ g being identity functions. 47. (i) If f : X → Y and g : Y → X are functions such that g ◦ f = 1 X , then f is injective and g is surjective. (ii) A function f : X → Y has an inverse g : Y → X if and only if f is a bijection. Proof.