Algebra

Books #

Algebra - Michael Artin
Abstract Algebra; An Introduction

Congrunce in \( \mathbb{Z} \) #

\( a \equiv c (mod n) \implies [a] = [c] \)
Let \( n > 1 \) be an integer and consider the congruence modulo n then it follows that if a is any integer and r is any remainder when a is divided by n then \( [a] = [r] \) furthermore there are n distinct congruence classes \( [0], [1],... [n-1] \)
The sum of two classes is the class containing both classes \( [a] \oplus [c] = [a + c] \)

Class Properties #

If \( [a] \in \mathbb{Z}_n ^ [b] \in \mathbb{Z}_n \implies [a] \oplus [b] \in \mathbb{Z}_n \)
\([a] \oplus ([b] \oplus [c]) = ([a] \oplus [b]) \oplus [c] \)
\( [a] \oplus [b] = [b] \oplus [a] \)
\( [a] \oplus [0] = [a] = [0] \oplus [a] \)
\( \forall [a] \in \mathbb{Z}_n \) the equation \( [a] \oplus X = [0] \) has a solution
If \( [a] \in \mathbb{Z}_n ^ [b] \in \mathbb{Z}_n \implies [a] \odot [b] \in \mathbb{Z}_n \)
\([a] \odot ([b] \odot [c]) = ([a] \odot [b]) \odot [c] \)
\( [a] \odot [b] = [b] \odot [a] \)
\( [a] \odot [1] = [a] = [1] \odot [a] \)
\( [a] \) is a unit in \( \mathbb{Z}_n \) iff \( (a,n) = 1 \in \mathbb{Z} \)

Rings #

A ring is a nonempty set R equipped with two operations usually written as addition and multiplication that satisfy the following:
\( a \in R \land b \in R \implies a + b \in R \)
a + (b+c) = (a+b)+ c
a+ b = b + a
There is \( 0_R \in R | a + 0_R = a = 0_R + a\)
\( \forall a \in R, a + x = 0_R\) has a solution
If \( a \in R ^ b \in R \implies ab \in R \)
a(bc) = ab(c)
a(b+c) = ab+ac ^ (a+b)c = ac+bc A ring without identity is known as a rng.
An Integral domain is a commutative ring R with identity \( 1_R \neq 0_R\) where given \( a,b\in R ab - 0_R \implies a = 0_R \lor b = 0_R\)
A field is a commutative ring R \( 1_R \neq 0_R\) where each \( a \neq 0_R \) the equation ax =1 has a solution in R
(r,s) + (r', s' ) = (r + r' , s + s') and (r,s)(r', s') = (rr', ss')
\( \mathbb{Z} \) is a subring of the ring \( \mathbb{Q} \) of rational numbers and \( \mathbb{Q} \) is a subring and subfield and of the field \( \mathbb{R} \) as \( \mathbb{Q} \) itself is a field. Simiarly \( \mathbb{R} \) is a subfield and subring of the field \( \mathbb{C} \)
Suppose that R is a ring and that S is subset of R then S is closed under addition and multiplication, it contains a zero element and given an a element in S, a + x = 0 has a solutuion that is in S. Thus S is a Subring.
Another definition is let S be a nonempty subset of a ring R if S is closed under subtraction and multiplication then S is a subring of R.
An integral domain contains no zero divisors.
Every field is an integral domain
Every finite integral domain R is a field
A ring R is isomorphic to a ring S if there exists a bijective function between the two and if \( f(a+b) = f(a) + f(b) \) and \( f(ab) = f(a)f(b) \)
Let R and S be rings then a function is said to be a homomorphism if \( f(a+b) = f(a) + f(b) \) and \( f(ab) = f(a)f(b) \)
If F: R to S is a homomorphism of rings then \( f(0_R) = 0_S \) , \(f(-a) = f(a) \forall a \in R \) , \( f(a-b) = f(a) - f(b) \forall a \in R \)
If R is a ring with identity and f is surjective then S is a ring with identity f(1) whenever u is a unit in R then f(u) is a unit in S and \( f(u)^{-1} = f(u^{-1}) \)
The property of being a commutative ring is preserved by isomorphism. Therefore no commutative ring can be isomorphic to a noncommutative ring.

Groups #

A group is a nonempty set G equipped with a binary operation * that satisfies the following axioms:

Closure
Associativity
There exists an element in a group G known as the identity such that \( a \dot e = a = e \dot a \)
There exists an inverse for a group st \( a * d = e \) and \( d * a = e\)
A group is abelian if it is commutative (a * b = b * a)
A group G has a unique identity element, a unique inverse, and cancellation holds.
if G is a group then \( (ab)^{-1} = b^{-1}a^{-1} \) and (a^{-1})^{-1} = a
A subset H of a group G is a subgroup of G if H itself a group under the operation in G
Let G and H be groups with the group operation by *. G is isomorphic to a group H if there exists a bijective function from G to H that also preserves the property \( f(a * b) = f(a) * f(b) \)
Let G and H be groups, a function is said to be a homomorphism if \( f(a * b) = f(a) * f(b) \)
Let K be a subgroup of a group G. Then the relation of congruence modulo K is reflexive, symmetric, and transitivity.
Let K be a subgroup of a group G then \( a \equiv c (mod K) \) iff \( Ka = kc\)
If G has a n order then G is isomorphic to \( \mathbb{Z}_2 \) this follows all the way to when G has an order of 7
a subgroup N of a group G is said to be normal if Na = aN for every a in G.
Let N be a normal subgroup of a group G then if Na = nc and Nb = Nd in G/N then Nab = Ncd.
Let N be a normal subgroup of a group G then G/N is a group under the operation defined by (Na)(Nc) = Nac
If G is finite then the order of G/N is |G|/|N|
If G is abelian group then so is G/N
The group G/N is called the quotient group or factor group of G by N.
Ka + Kb = K(a+b) is the same as [a]+[b] = [a + b]
Let \(f: G \rightarrow H\) be a homomorphism of groups. Then the kernel of F is the set \( \)
Let \(f: G \rightarrow H \) be a homomorphism of groups with kernel K. Then K is normal subgroup of G.
The kernel of a homomorphism f measures how far f is from being injective.
Let \(f: G \rightarrow H \) be a group homomorphism with kernel K. Then f(a) = f(b) iff Ka = Kb.
-Let \( f: G \rightarrow H \) be a surjective homomorphism of groups with kernel K. Then the quotient group G/K is isomorphic to H.
Let G be a group and let N be a subgroup of G. Therefore \( \forall s \in G \) and \( \forall a \in N \) , \( s^{-1}as \in N \) .