Basic Ring - Ring (R, +, . ) (2 binary operations): (R,...

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Unformatted text preview: Ring (R, +, . ) (2 binary operations): (R, +) is an abelian group . is associative + . is distributive (left & right) a . 1 = 1 . a = a Ring with unity Commutative Ring No zero-divisors a . b = b . a 1 ≠ 0 except the zero ring {0} (R*, . ) is a group Division Ring Integral Domain (R*, . ) is an abelian group Field Summands notation in Rings Given an element ‘a’ in ring R : • n . a = a+a+…+a (n of them) when n>0 • 0 . a = 0 in R, when n = integer zero • n . a = (-a) + …+ (-a) (n of them) when n<0 Basic identities Additive identity 0 in R, any a, b in R : 1. 0a = a0 = 0 2. a(-b) = (-a)b = -(ab) (note the order of ab) 3. (-a)(-b) = ab 4. (n . 1) (m . 1) = (nm) . 1 Ring Homomorphism & Isomorphism • φ : R → R’ is a homo if for any a, b in R 1. φ (a+b) = φ (a)+’ φ (b) 2. φ (ab) = φ (a) φ (b) (Ring homo implies Group homo) • φ : R → R’ is a isomo if 1. φ is homo 2. φ is 1-1 & onto Basic homomorphism theorem φ : R → R’ is a homo, (...
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Basic Ring - Ring (R, +, . ) (2 binary operations): (R,...

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