Abstract algebra

Rings

Ring axioms

A ring is a nonempty set $R$ equipped with two operations (usually written as addition and multiplication) that satisfy the following axioms. For all $a,b,c\in R$:

1. (Closure under addition) $a+b\in R$
2. $a+(b+c)=(a+b)+c$
   (Associative addition)
3. $a+b=b+a$
   (Commutative addition)
4. There is an element $0_R\in R$ such that $a+0_R=a=0_R+a$
   (Additive identity/zero element)
5. The equation $a+x=0_R$ has a solution in $R$
   (Additive inverse)
6. $ab\in R$
   (Closure under multiplication)
7. $a(bc)=(ab)c$
   (Associative multiplication)
8. $a(b+c)=ab+bc$ and $(a+b)c=ac+bc$
   (Distributive multiplication)

Commutative ring

A commutative ring is a ring $R$ that satisfies:

9. $ab=ba$ for all $a,b\in R$.
   (Commutative multiplication)

Ring with unity/identity

A ring with unity/identity is a ring $R$ that contains an element $1_R$ satisfying:

10. $a 1_R=a=1_R a$ for all $a\in R$.
    (Multiplicative identity/identity element)

Integral domain

An integral domain is a commutative ring $R$ with identity $1_R\neq 0_R$ that satisfies:

11. Whenever $a,b\in R$ and $ab=0_R$, then $a=0_R$ or $b=0_R$.

Fields

A field is a commutative ring $R$ with identity $1_R\neq0_R$ that satisfies:

12. For each $a\neq 0_R$ in $R$, the equation $ax=1_R$ has a solution in $R$.

Cartesian product of rings

Let $R$ and $S$ be rings. Define addition and multiplication on the Cartesian product $R\times S$ by:

$(r,s)+(r',s')=(r+r',s+s')$ and $(r,s)(r',s')=(rr',ss')$

Then $R\times S$ is a ring. If $R$ and $S$ are both commutative, then so is $R\times S$. If both $R$ and $S$ have an identity (additive and multiplicative?), then so does $R\times S$.

Subrings

Suppose that $R$ is a ring and that $S$ is a subset of $R$ such that

1. $S$ is closed under addition: if $a,b\in S$, then $a+b\in S$,
2. $S$ is closed under multiplication: if $a,b\in S$, then $ab\in S$,
3. $0_R\in S$,
4. if $a\in S$, then the solution of the equation $a+x=0_R$ is in $S$.

Units and zero divisors

An element $a$ in a ring $R$ with identity is called a unit if there exists $u\in R$ such that $au=1_R=ua$. In this case the element $u$ is called the (multiplicative) inverse of $a$ and is denoted $a^{-1}$.

An element $a$ in a ring $R$ is a zero divisor provided that:

1. $a\neq 0_R$.
2. There exists a nonzero element $c$ in $R$ such that $ac=0_R$ or $ca=0_R$

Ring properties

Cancelation in addition

If $a+b=a+c$ in a ring $R$, then $b=c$.

Zero product, distributivity

For any elements $a$ and $b$ of a ring $R$:

1. $a\cdot 0_R=0_R=0_R\cdot a$. In particular, $0_R\cdot 0_R=0_R$.
2. $a(-b)=-ab$ and $(-a)b=-ab$.
3. $-(-a)=a$.
4. $-(a+b)=(-a)+(-b)$.
5. $-(a-b)=-a+b$.
6. $(-a)(-b)=ab$.
7. If $R$ has an identity, then $(-1_R)a=-a$.

Subrings

Let $S$ be a nonempty subset of a ring $R$ such that

1. $S$ is closed under subtraction (if $a,b\in S$, then $a-b\in S$);
2. $S$ is closed under multiplication (if $a,b\in S$, then $ab\in S$).

Then $S$ is a subring of $R$.

Cancelation in multiplication

Cancellation is valid in any integral domain $R$: If $a\neq 0_R$ and $ab=ac$ in $R$, then $b=c$.

Fields as integral domains

Every field $F$ is an integral domain.

Finite integral domains as fields

Every finite integral domain $R$ is a field.

Isomorphisms & Homomorphisms

To summarize terms:

Injective | All domain values mapped Non-injective | Not all domain values mapped Surjective | All co-domain values represented Non-surjective | Not all co-domain values represented Bijective | Both injective and surjective

Isomorphisms

Suppose $R$ and $S$ are rings and $\phi$ is a function that maps $R$ to $S$ ($\phi:R\rightarrow S$ ). $\phi$ is an isomorphism if for all $a,b\in R$:

1. $\phi$ is injective,
2. $\phi$ is surjective,
3. $\phi(a+b)=\phi(a)+\phi(b)$ and $\phi(a\cdot b)=\phi(a)\cdot\phi(b)$ for all $a,b\in R$.

Another way of stating bijection would be:

1. Distinct elements of $R$ must get distinct new labels
   If $r\neq r'$ in $R$, then $\phi(r)\neq\phi(r')$ in $S$

2. Every element of $S$ must be the label of some element in $R$
   For each $s\in S$, there is an $r\in R$ such that $\phi(r)=s$

Intuitively for a function to be bijective an inverse isomorphic function must also exist.

Homomorphisms

However many functions exists which satisfy the third condition but are not bijective. These can be called homomorphisms. Thus every isomorphism is really a homomorphism with bijection.

Properties of homomorphisms

Letting $\phi:R\rightarrow S$ be a homomorphism of rings, then

1. $\phi(0_R)=0_S$,
2. $\phi(-a)=-\phi(a)$ for every $a\in R$,
3. $\phi(a-b)=\phi(a)-\phi(b)$ for all $a,b\in R$.

If $R$ is a ring with identity and $\phi$ is surjective, then

4. $S$ is a ring with identity $1_S=\phi(1_R)$,
5. Whenever $u$ is a unit in $R$, then $\phi(u)$ is a unit in $S$ and $\phi(u)^{-1}=\phi(u^{-1})$.

Polynomial rings

Fix $F[x]$ a field for this section.

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{% include tooltip.html text="Definition ?" content=" $a(x)\in F[x]$ is irreducible if $\deg(a(x))>0$ (positive) and if whenever $a(x)=b(x)c(x)$ then $b(x)$ or $c(x)$ is a unit. " %}

{% include tooltip.html text="Theorem 4.12" content=" The following are equivalent:

1. $a(x)\in F[x]$ is irreducible,
2. Whenever $a(x)\mid b(x)c(x)$ either $a(x)\mid b(x)$ or $a\mid c(x)$,
3. We cannot write $a(x)=b(x)c(x)$ where $b(x)$ and $c(x)$ both have positive degree. " %}

Uniqueness of factorization

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$$ a(x)=p_1(x)p_2(x)\cdots p_n(x) $$

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Groups