website/content/blog/abstractdef.md

title	date	draft	tags	math
Useful Abstract Definitions	2019-10-27T23:49:57-04:00	false	Math	true

Chapter 17

By a ring we mean a set A with operations called addition and multiplication which satisfy the following axioms:

• A with addition alone is an abelian group
• Multiplication is associative
• Multiplication is distributive over addition

Since \langle A, + \rangle is an abelian group, there is in A a neutral element for addition. This is called the zero element. Also, every element has an additive inverse called its negative.

If multiplication in a ring A is commutative then we say that A is a commutative ring.

If a ring A has a neutral element for multiplication then we say that the neutral element is the unity of A.

If A is a ring with unity, there may be elements in A which have a multiplicative inverse. Such elements are said to be invertible.

If A is a commutative ring with unity in which every nonzero element is invertible, then A is called a field.

In any ring, a nonzero element a is called a divisor of zero if there is a nonzero element b in a ring such that the product ab or ba is equal to zero.

An integral domain is defined to be a commutative ring with unity that has the cancellation property. Another way of saying this is that an integral domain is a commutative ring with unity and has no zero divisors.

Chapter 18

Let A be a ring, and B be a nonempty subset of A. B is called a subring if it satisfies the following properties:

• Closed with respect to addition
• Closed with respect to negatives
• Closed with respect to multiplication

Let B be a subring of A. We call B an ideal of A if xb, bx \in B for all b \in B and x \in A.

The principle ideal generated by $a$, denoted \langle a \rangle is the subring defined by fixing an element a in a subring B of A and multiplying all elements of A by a. $$ \langle a \rangle = { xa : x \in A } $$ A homomorphism from a ring A to a ring B is a function f : A \to B satisfying the identities: $$ f(x_1 + x_2 ) = f(x_1) + f(x_2) \ f(x_1x_2) = f(x_1)f(x_2) $$ If there is a homomorphism from A onto B, we call B a homomorphic image of A.

If f is a homomorphism from a ring A to a ring B, the kernel of f is the set of all the elements of A which are carried by f onto the zero element of B. In symbols, the kernel of f is the set $$ K = {x \in A: f(x) = 0} $$ If A and b are rings, an isomorphism from A to B is a homomorphism which is a one-to-one correspondence from A to B. In other words, it is injective and surjective homomorphism.

If there is an isomorphism from A to B we say that A is isomorphic to B, and this fact is expressed by writing A \cong B.

Chapter 19

Let A be a ring, and J an ideal of A. For any element a \in A, the symbol J + a denotes the set of all sums j + a as a remains fixed and j ranges over J. That is, $$ J + a = {j + a : j \in J} $$ J + a is called a coset of J in A.

Now think of the set which contains all cosets of J in A. This set is conventionally denoted by A / J and reads A mod J. Then, A / J with coset addition and multiplication is a ring.

An ideal J of a commutative ring is said to be a prime ideal if for any two elements a and b in the ring, $$ ab \in J \implies a \in J \text{ or } b \in J $$ Theorem: If J is a prime ideal of a community ring with unity A, then the quotient ring A / J is an integral domain.

An ideal J of A with J \ne A is called a maximal ideal if there exists no proper ideal K of A such that J \subseteq K with J \ne K.

Theorem: If A is a commutative ring with unity, then J is a maximal ideal of A iff A/J is a field.