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Abstract Algebra
How to prove Eculidean algorithm
Jun 26th
a, b in Z, and p is prime
if p | ab
=> p | a or p | b
proof by contradiction:
assume p is not divided by a and p is not divided by b
p | ab => gcd(p, a) = 1 => xp + ya = 1 such that there exists x, y in Z(Eculidan algorithm)
since xp + ya = 1 => b(xp + ya) = b => bxp + bya = b
since a | ab = > p | bya and p | bxp
=> p | (bxp + bya) => p | b
that contradict to our assumption
how to prove a ring is subring?
Jun 11th
example:
how to prove 3Z is subring of Z (Z is integer)
we only need to prove 3Z is closed under subtraction and multiplication.
subtraction => 0 and additive inverse in the 3Z.
let a, b in 3Z.
if a – b in 3Z then
let a = b, => b + (-b) = in 3Z
=> 0 in 3Z and additive inverse -b in 3Z
Existence of Factor Rings
Jun 11th
let R be the ring, and A be the subring of R. The set of cosets
{r + A | r in R} is a ring under the operations
(r + A) + (s + A) = r + s + A and
(r + A)*(s + A) = rs + A if and only if A is an ideal of R
Abstract Algebra: what is Ideal?
May 30th
Ideal is defined by:
if R is a Ring and I is subring of R, then
if r in I, and x in R then
r*x in I and x*r both in I
example of Ideal:
let R = Z, I = 2Z
r in Z
x in 2Z
then r*x = z*2Z = 2*Z
x*r = 2Z*Z = 2*Z
hence 2Z is an Ideal
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abstract algebra: what is subring?
May 30th
subring defined by:
if R is a ring,
1) I is subset of R
2) I is also a ring
example:
let R = Z (integer)
I = 2Z is subring of R, since 2Z is also a ring,
is (2Z + 1) also a subring of R?, no
proof:
let a, b in Z
=> 2*a + 1 in 2Z + 1
=> 2*b + 1 in 2z + 1
let (2*a + 1 ) + (2*b + 1)
=> 2*a + 2*b + 1 + 1
=> 2(a + b) + 2
=> 2(a + b + 1)
=> 2(a + b + 1) is not in 2Z + 1
hence 2(a + b + 1) is not closed under addition
so 2Z + 1 is not a ring,
abstract algebra: what is ring?
May 30th
ring is defined by
1) two operator: addition(+) and multiplication(*)
if a, b in R, then a + b in R
if a, b in R, then a*b in R
2) there exists 0 such that if a in R, then a + 0 = a
there exists additive inverse -a such that if a in R, then a + (-a) = 0examples of ring:
Integer, real number, complex number.
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