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111B Def and Thm-1 - Accumulation of Denitions Theorems...

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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 non-zero elements, than R is a Division Ring , i.e. ( R ) = R \{ 0 } 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 non-zero 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 zero-divisor 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
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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 zero-divisors. [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. [2.10] Proposition : If R is a unitary ring then char ( R ) = min { n N | ( n )(1) = 0 } if it exists and char ( R ) = 0 otherwise. [2.11] Proposition : The characteristic of an integral domain R is either a prime or 0. Chapter 3: Fermat’s and Euler’s Thereom [3.3] Theorem (Euler) : Let 1 < n N and let a Z with gcd ( a, n ) = 1. Then a ϕ ( a ) 1 modn .
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