Study Guide Midterm 1

Study Guide Midterm 1 - Section 1 – Rings: First...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ξψ-1 ( 29 γ ψ ( 29 ψ Γ Then RG, with this addition and multiplication is a ring called group-ring 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 ....
View Full Document

This note was uploaded on 04/08/2008 for the course MATH 4041 taught by Professor Gallagher during the Spring '08 term at Columbia.

Page1 / 6

Study Guide Midterm 1 - Section 1 – Rings: First...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online