This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Accumulation of Definitions, Theorems, Corollaries, and Propositions provided by Professor Robert Boltje in 111B: Ring Theory, Spring 2010. If any errors are found, contact Gary Kirby at [email protected] Chapter 1: Rings and Fields [1.1] Definition : A Ring ( R, + , × ) is a set R with two binary opertaions called addition and multiplication s.t. 1. ( R, +) is an abelian group 2. ( R, × ) is associative 3. (Distributive Laws) (a) a ( b + c ) = ( ab ) + ( ac ) (b) ( a + b ) c = ( ac ) + ( bc ) Types/Cases of Rings : 1. If ( R, + , × ) is multiplicatively commutative, than R is a Commutative ring 2. If ( R, + , × ) has a multiplicative identity, 1 R , than R is a Unitary Ring 3. If ( R, + , × ) has a multiplicative inverse for all nonzero elements, than R is a Division Ring , i.e. ∪ ( R ) = R \{ } 4. If ( R, + , × )is a Division Ring, that is commutative, than R is a Field 5. If ( R, + , × ) has a multiplicative identity with 0 R = 1 R , no 0 divisors, and commutative, than R is a Integral Domain 6. If ( R, + , × ) is a Integral Domain, and every nonzero element is a unit, than R is a Field [1.8] Definition : Let R and S be rings. A function ϕ : R→ S is called a ring homomorphism if ϕ ( a + b ) = ϕ ( a ) + ϕ ( b ) and ϕ ( ab ) = ϕ ( a ) ϕ ( b ) ∀ a,b ∈ R. If R and S are unitary rings and if additionally ϕ (1 R ) = 1 S , than ϕ is a unitary ring homomorphism . A homomorphism ϕ : R→ S is called an isomorphism if ϕ is bijective, thus R ∼ = S. [1.11] Definition : Let R be a ring. A subset S ≤ R is called a subring of R if S is a subgroup under addition and if S is closed under multiplication. In this case, S is again a ring with induced addition and multplication. If R is unitary and 1 R ∈ S, then 1 R is a unity in S and called a unitary subring. Let F be a field. A subring E of F is again a field with the induced addition and multiplication is than called a subfrield of F. (Automatically 1 E = 1 R ) Chapter 2: Integral Domains [2.2] Definition : A nonzero element of a ring R, is called a zerodivisor if there exists an element b ∈ R with ab = 0 or ba = 0. [2.3] Proposition : Let 1 < n ∈ N and let a ∈ { 1 ,...,n 1 } 1 1. a is a zero divisor in Z n ⇔ gcd ( a,n ) > 1 2. a is a unit in Z n ⇔ gcd ( a,n ) = 1 [2.4] Definition : We say that the cancellation laws hold in a ring R if ∀ a,b ∈ R with a = 0 has: ab = ac ⇒ b = c and ba = ca ⇒ b = c. [2.5] Proposition : The cancellation laws hold in a ring R ⇔ R has no zerodivisors. [2.8] Proposition : Let R be a finite ring. R is a field ⇔ R is an integral domain. [2.9] Definition : Let R be a ring, If there exists n ∈ N with na = 0 ∀ a ∈ R then the smallest such n is called the characterisic of R, char ( R ). Otherwise, one sets char ( R ) = 0....
View
Full
Document
This note was uploaded on 12/16/2010 for the course MATH 100 taught by Professor Jim during the Spring '10 term at American Jewish University.
 Spring '10
 jim

Click to edit the document details