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Daryl

on Jan 27, 2009

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abstract algebra - ring

abstract algebra - ring

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• Definition of Rings and Examples By: John Wilson Calalo and Daryl Sacay
• A ring is a none empty set R equipped with two operations that satisfy the following axioms. For all a, b, c, є R:
• 1. If a є R and b є R, then a+b є R [closure for addition] 2. a + (b + c) = (a + b) + c [associative addition] 3. a + b = b + a [commutative addition] 4. There is an element O R in R such that a + O R = a O R + a for every a є R [additive identity or zero element]
• 5. For each a є R, the equation a + x = O R has a solution in R. 6. If a є R and b є R, then ab є R. [closure for multiplication] 7. A (bc) = (ab) c [associative multiplication] 8. A (b + c) = ab + ac and (a + b) c = ac + bc [distributive laws]
• Commutative Ring is a ring R that satisfies this axiom: • ab = ba for all a, b [commutative multiplication]
• Example: The set of integers Z, with the usual addition and multiplication, is a commutative ring with identity.
• Let E be the set of even integers with the usual addition and multiplication
• The set of odd integers with the usual addition and multiplication is not a ring.
• The set T= {r, s, t, z} equipped w/ the addition and multiplication defined by the following tables is a ring.
• z r s t r z t s s t z t t s r z z r s t z r s t +
• z z z z z z r r z z s s z z t t z r s t z r s t •
• Let  (R) be the set of all 2x2 matrices over the real numbers, that is,  (R) consist of all arrays a b c d
• Two matrices are equal provided that the entries in corresponding positions are equal; that is, a b c d = r t s u If and only if a = r, b = s, c = t, d = u
• For example, 4 0 2+2 0 but 1 3 ≠ 3 5 -3 1 1-4 1 5 2 1 2
• Multiplication of Matrices is defined by: a b c d = w x y z aw=by ax=bz cw=dy cx=dz
• For example,
• 3 1 -5 2.1+3.6 2(-5) +3.7
• 0 -4 6 7 0.1=(-4)6 0(-5)+(-4)(7)
• The multiplicative identity element is the matrix I = 1 0 0 1
• For instance a b 1 0 = a.1+b.0 a.0+b.1 c d 0 1 c.1+d.0 c.0+d.1 = a b c d
• Nonzero elements maybe thezero element;
• 6 -3 -9 = 4(-3)+6.2 4(-9)+6.6
• 2 3 2 6 2(-3)+3.2 2(-9)+3.6
= 0 0 0 0