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| title | date | draft | tags | math | |
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| Useful Abstract Definitions | 2019-10-27T23:49:57-04:00 | false |
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Chapter 17
By a ring we mean a set A with operations called addition and multiplication which satisfy the following axioms:
Awith 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.