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...
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 fo...
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 a...
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
<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) <...
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
<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 ...
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Daryl

  1. 1. Definition of Rings and Examples By: John Wilson Calalo and Daryl Sacay
  2. 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. 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. 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. 5. Commutative Ring is a ring R that satisfies this axiom: • ab = ba for all a, b [commutative multiplication]
  6. 6. Example: The set of integers Z, with the usual addition and multiplication, is a commutative ring with identity.
  7. 7. Let E be the set of even integers with the usual addition and multiplication
  8. 8. The set of odd integers with the usual addition and multiplication is not a ring.
  9. 9. The set T= {r, s, t, z} equipped w/ the addition and multiplication defined by the following tables is a ring.
  10. 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. 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. 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. 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. 14. For example, 4 0 2+2 0 but 1 3 ≠ 3 5 -3 1 1-4 1 5 2 1 2
  15. 15. Multiplication of Matrices is defined by: a b c d = w x y z aw=by ax=bz cw=dy cx=dz
  16. 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. 17. The multiplicative identity element is the matrix I = 1 0 0 1
  18. 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. 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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