Commutative Algebra Ongoing BC

Question 1

defn. commutative ring?

Answer 1

A set R equipped with two operations + & x such that they satisfy the eight axioms.

• addition is associative.
• addition is commutative.
• a∈R ⇒ -a∈R.
• 0∈R ⇒ a+0 = a.
• 1∈R ⇒ a x 1 = a.
• multiplication is associative.
• a(b + c) = ab +ac. distributive
• multiplication is commutative.

Question 2

axioms?

Answer 2

(R,+) is an abelian group
a(b+c) ⇒ ab+bc
ab=ba

Question 3

defn. Inverse

Answer 3

R is a commutative ring and x∈R, an element x¯¹∈R is an inverse of x if xx¯¹ =1

Question 4

properties of inverses

Answer 4

they are unique,

inverse of 0 doesn’t exist

Question 5

Defn. unit

Answer 5

x∈R is a unit if it is an invertible element of R.

Question 6

Defn. Abelian Group

Answer 6

a commutative group

Question 7

Defn. U(R)?

Answer 7

U(R)= {x∈R | x is a unit}

group of invertible elements

Question 8

Defn. Field ?

Answer 8

a field is a commutative ring K such that U(K) =K{0}

ie, a field is a ring where all non-zero elements are invertible

Question 9

Defn. Domain ?

Answer 9

a domain is R such that a,b∈R, ab=0 ⇒ a=0 or b=0

Question 10

relationship between feild and domain ?

Answer 10

a field is always a domain.

ie a group of invertible elements excluding zero is always a domain

Question 11

defn. Monomials

Answer 11

monomials in n variables are in one-to-one correspondence with the set ℕ^n.
ℕ^n = {α = (k_1,… , k_n) | k_1,… ,k_n ∈ℕ} the set of all n-tuples of natural numbers.

Question 12

multiindex

Answer 12

an element (α) of a set ℕ^n

Question 13

M(X_1,…,X_n)

Answer 13

the set of all monomials in n variables

Question 14

addition and multiplication of monomials

Answer 14

addition is not defined

multiplication is defined as X³X² = X(³+²)

Question 15

total degree of a monomial

Answer 15

= the sum of the powers

Question 16

defn. Poset (partially ordered set)

Answer 16

(S,≼) is a partially ordered set if S is a set and ≼ is a binary relation on S which is

• reflexsive, x∈S ⇒ x≼x
• antisymmetric: x≼y, y≼x ⇒ x=y
• transitive: x≼y, y≼z ⇒ x≼z

Question 17

Lecture 3

Answer 17

Lecture 3

Question 18

Dicksons Lemma

Answer 18

S is a sub set of M(X_1,…,X_n): then # Smin

Question 19

The partial order | on M(X_1,…,X_n)

Answer 19

take a picture

Question 20

Dickson’s Lemma

Answer 20

Let S be a subset of (M(X_1, . . . ,X_n), |). Then.

1. S_min is finite.
2. Every element of S is divisible by at least one element of S_min.

Question 21

Minimal element

Answer 21

A minimal element of a partially ordered set (S,≼) is x∈S such that there is no y∈S with y ≺ x.
Denote by S_min the set of minimal elements of S.

Question 22

Dickson’s Lemma

Answer 22

Let S be a subset of (M(X_1, . . . ,X_n), |). Then:

1. S_min is finite.
2. Every element of S is divisible by at least one element of S_min.

Question 23

Monomial orderings

Answer 23

A monomial ordering is a partial order ≼ on M = M(X_1,…,X_n) which satisfies:

• ≼ is a total order. (i.e m,m’∈M ⇒ m≼m’ or m≻m’;
• m∈M ⇒ 1≼m
• If m’≼m then, for every m_1 ∈M, m_1m’ ≼ m1m.

Question 24

Lex Ordering

Answer 24

Variables are ordered first X_1 ≻ X_2≻…X_n

X^a ≻ X^b if a - b > 0.

Question 25

DegLex Ordering

Answer 25

Variables are ordered first X_1 ≻ X_2≻…X_n
totaldeg(X^a) > totaldeg(X^b)
then use Lex if the total degrees are equal.