Defn's/Statements Flashcards
Define a Noetherian Ring
R is noetherian, if every ideal of R is finitely generated
What is a Euclidean Ring?
A ring equipped with a Euclidean Norm
What is Hilberts Basis Theorem?
If K is noetherian therefore K[X] is a noetherian ring
What is a Euclidean Norm?
A euclidean norm on a ring R is a function A : R → N such that for all a, b ∈ R{0}, there exists q ∈ R such that a = qb or N(a − qb)
What is meant by saying that b is an associate of a?
b = xa where x is a unit of R
What is meant by saying that a is irreducible in R?
a is not 0, not a unit and not a product of two non-units
State Dickson’s Lemma about minimal monomials of S.
Let S be a subset of (M(X1, . . . ,Xn), |). Then:
- Smin is finite.
- Every element of S is divisible by at least one element of Smin.
billy. Define a Prime in a domain
a non-unit p is called a prime if p/ab ⇒ p/a or p/b
Define a monomial ordering and it’s conditions.
A monomial ordering is a partial order ≤ on M = M(X1, . . . ,Xn) which satisfies:
1) ≤ is a total order i.e. m, m’ ∈ M implies m ≤ m’ or m > m’
2) m ∈ M implies 1 ≤ m
3) if m’ ≤ m then for every m1 ∈ M , m1m’ ≤ m1m
What is meant by a prime in a commutative domain R?
p ∈ R is a prime if p is not a unit and ∀a, b ∈ R, p∤a, p∤b ⇒ p∤ab.
What is a unique factorisation domain (UFD)?
A domain where every non-unit is a product of primes
What is meant by saying that an ideal I of a commutative ring R is a maximal ideal?
I ≠ R and there exists no such J: I ⊊ J ⊊ R
What is meant by the radical, √I, of an ideal I?
√I = {a ∈ R | ∃n ∈ N : a^n ∈ I} √I is an ideal of R containing I
What is a radical ideal?
I is a radical ideal if √I = I
What is meant by the variety V(I) of an ideal I of K[X1, . . . , Xn]?
V(I) = {(a1, . . . , an) ∈ K^n | f(a1, . . . , an) = 0 ∀f ∈ I};
What is meant by the ideal ℑ(S) of a set S ⊆ K^n?
I(S) = {f ∈ K[X1, . . . , Xn] | f(a1, . . . , an) = 0 ∀(a1, . . . , an) ∈ S}
State the weak Nullstellensatz
If K is algebraically closed and I is a proper ideal of K[X1, . . ,Xn], then V(I) ≠ ∅.
If R is a UFD, what can this tell us
R[X] and R[X,Y,..] are also UFDs
Explain what is meant by saying that R satisfies the ascending chain condition (acc) on ideals.
Every ascending chain I1 ⊆ I2 ⊆ . . . of ideals of R eventually stabilises, that is, there exists n0: In0 = In for all n ≥ n0.
If D is a commutative domain, define the notion of an irreducible element of D
r ∈ D is irreducible if r is not a unit, and in every factorisation r = st with s, t ∈ D, one of
s, t is a unit.
Describe, without proof, all the irreducible elements of C[X].
{aX + b : a, b ∈ C, a does not equal 0} (or: polynomials of degree 1).
Give the definition of a Groebner basis with respect to monomial ordering of an ideal of R
A Groebner basis of an ideal I is a finite subset {f1, f2, . . . , fm} of I \ {0} such lm≼( fi ) generate the ideal LT(I) of leading terms of I.
What is √(√I) ?
√I
What is meant by saying that I is a prime ideal?
I is a proper ideal of R such that a, b ∈ R\I ⇒ ab∈R\I
ie
Prime Ideal = {ar | a∈I, r∈R/I}
