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abstract algebra - ring

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- 1. Definition of Rings and Examples By: John Wilson Calalo and Daryl Sacay
- 2. A ring is a none empty set R equipped with two operations that satisfy the following axioms. For all a, b, c, є R:
- 3. 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]
- 4. 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]
- 5. Commutative Ring is a ring R that satisfies this axiom: • ab = ba for all a, b [commutative multiplication]
- 6. Example: The set of integers Z, with the usual addition and multiplication, is a commutative ring with identity.
- 7. Let E be the set of even integers with the usual addition and multiplication
- 8. The set of odd integers with the usual addition and multiplication is not a ring.
- 9. The set T= {r, s, t, z} equipped w/ the addition and multiplication defined by the following tables is a ring.
- 10. z r s t r z t s s t z t t s r z z r s t z r s t +
- 11. z z z z z z r r z z s s z z t t z r s t z r s t •
- 12. Let (R) be the set of all 2x2 matrices over the real numbers, that is, (R) consist of all arrays a b c d
- 13. 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
- 14. For example, 4 0 2+2 0 but 1 3 ≠ 3 5 -3 1 1-4 1 5 2 1 2
- 15. Multiplication of Matrices is defined by: a b c d = w x y z aw=by ax=bz cw=dy cx=dz
- 16. <ul><li>For example, </li></ul><ul><li>3 1 -5 2.1+3.6 2(-5) +3.7 </li></ul><ul><li>0 -4 6 7 0.1=(-4)6 0(-5)+(-4)(7) </li></ul>
- 17. The multiplicative identity element is the matrix I = 1 0 0 1
- 18. 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
- 19. <ul><li>Nonzero elements maybe thezero element; </li></ul><ul><li>6 -3 -9 = 4(-3)+6.2 4(-9)+6.6 </li></ul><ul><li>2 3 2 6 2(-3)+3.2 2(-9)+3.6 </li></ul>= 0 0 0 0

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