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Unformatted text preview: Section 1 – Rings: First properties and Examples Definitions A monoid is a set M with an associative “multiplication,” and with an identity element, while a group is a monoid in which each element has an inverse. A ring is a set R, together with two maps R Ρ Ρ : 1. An addition a , b a a + β , making R into a commutative group 2. A multiplication a , b a ab , making R into a monoid. It is also require that multiplication is both left and right distributive over addition: a b + χ ( 29 = αβ + αχ β + χ ( 29 α = βα + χα 2200 α , β , χ Ρ Commutative rings are those in which the multiplication is also commutative. We say S is a subring of R if S is an additive subgroup of R and a multiplicative submonoid of R – meaning that s , σ Σ σσ Σ & 1 Σ , where 1 is the multiplicative identity of R. Let G be any multiplicative finite group, not necessarily abelian and let R be any ring. Denote RG the set of all maps from G to R. For f , g ΡΓ , put f + γ ( 29 ξ ( 29 = φ ξ ( 29 + γ ξ ( 29 & f γ ( 29 ξ ( 29 = φ ξψ1 ( 29 γ ψ ( 29 ψ Γ Then RG, with this addition and multiplication is a ring called groupring of G over R) with the constant 0 as 0 and the delta function defined by δ x ( ) = 1 for x = 1 for x 1 as the multiplicative identity. Let A be any additive abelian group. An endomorphism of A is any homomorphism ϕ : A A . Denote End(A) the set of all endomorphisms of A. Theorems A. Let A be any additive abelian group. With the sum defined as ϕ + Ψ ( 29 x ( ) = j x ( ) + Y x ( ) and product defined as (composite) ϕ οΨ ( 29 x ( ) = j Y x ( ) ( ) , End A ( 29 is a ring. Section 2 – Homomorphisms of Rings, Ideals, Factor Rings Definitions: Let R and S be rings. A map ϕ : R S is a (ring) homomorphism if the following two conditions are satisfied: 1. ϕ is a homomorphism of the additive group R , + { } 2. ϕ is a homomorphism of the multiplicative monoid R ,g { } meaning a , b Ρ ϕ αβ ( 29 = ϕ α ( 29 ϕ β ( 29 and ϕ 1 ( ) = 1 . A subset M of a ring is an ideal if the following conditions are satisfied: 1. M is a subgroup of the additive group R , + { } 2. RM Μ meaning a Ρ & β Μ αβ Μ ....
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This note was uploaded on 04/08/2008 for the course MATH 4041 taught by Professor Gallagher during the Spring '08 term at Columbia.
 Spring '08
 Gallagher
 Algebra, Multiplication

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