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}
What is a monomial ideal of K[X1, . . . , Xn]? Explain why the ideal {0} is a monomial ideal.
A monomial ideal of K[X1, . . . , Xn] is an ideal generated by a set of monomials in X1, . . . , Xn. The ideal {0} is a monomial ideal because it is generated by the empty set.
Define LT≼( I ), the ideal of leading terms of I.
LT≼( I ) is the ideal generated by the leading monomials of the non-zero polynomials in I.
What is a subring of a domain?
A domain
What is a reduced Groebner basis?
A reduced Gr¨obner basis is a Gr¨obner basis {g1, . . . , gn} such that all the gi are monic polynomials and for any i ≠ j, the polynomial gi is reduced with respect to gj .
What is meant by saying that a commutative ring R is a principal ideal domain?
R is a domain and every ideal of R can be generated by one element.
Define the notion of a nilpotent element of R.
An element a ∈ R is nilpotent if there is n ∈ N such that a^n = 0.
What is it to say that a field K is algebraically closed?
We say that a field K is algebraically closed if every non-constant polynomial in K[X] has a root in K
State without proof Hilbert’s Basis Theorem for polynomials over a field.
If K is a field then K{X1,…Xn] is a noetherian ring
A Euclidean ring is also another type of ring, what is this ring?
A principal ideal ring
A principal ideal domain is what?
A domain is a domain whose ideals are generated by one element ( i.e. a PIR which is also a domain)
A principal ideal domain (PIR which is a domain) is also another type of domain, what is this domain?
A unique factorisation domain (UFD)
if K is a field, K[X] is a what type of domain?
A unique factorisation domain (UFD)
We say that a polynomial f = anX^n + . . . + a1X + a0 ∈ R[X] is Eisenstein, if there is a prime p ∈ R such that …?
1) p ∤ an
2) p | an-1, an-2, ….., a1, a0
3) p^2 ∤ a0
What is the The Eisenstein Criterion?
If R is a UFD and f ∈ R[X] is Eisenstein, then f is irreducible in (ℱr R)[X].
what is the nilradical Nil(R) of a ring R?
Nil(R) = √
Every field R is a type of domain, what is this domain?
A unique factorisation domain (UFD)
What is meant by saying that a polynomial f ∈ Q[X, Y, Z] is reduced modulo a set F ⊂ Q[X, Y, Z] \ {0} with respect to a monomial ordering ≼ ?
No monomial in f is divisible by the ≼-leading term of any of the elements of F
Describe, without proof, all irreducible polynomials in R[X]
aX + b with a not equal to 0 and aX^2 + bX + c with b^2 − 4ac
irreducible polynomials in Q[X] when degree ≥ 2?
are polynomials with no root in Q.
to find this out,
multiply up to get integer coefficents, and roots are of the form p/q where p | a_0 and q | a_n
In a UFD, what’s the relationship with primes and irreducible elements
Irreducible elements implies prime
Prime element implies irreducible, except 0.
What is meant by a remainder of a polynomial f ∈ Q[X, Y, Z] modulo a set F ⊂ Q[X, Y, Z] with respect to a monomial ordering ≼ ?
A polynomial r ∈ Q[X, Y, Z] such that f can be reduced to r modulo F with respect to ≼ , and r is reduced modulo F.
unit x unit = ? unit x non unit = ?
unit x unit = unit, unit x non unit = non unit
solution of the membership problem for monomial ideals
let T⊂M[X] be a set of monomials. A polynomial f belongs to the ideal of K[X] every monomial in f is divisible by at least one monomial from T.
if f mod G has a remainder, then..?
r is unique, as G is a groebner basis
list the primes in ℝ
0 mate
list the primes in ℝ[X]
since all primes are irreducible elements.
aX + b with a not equal to 0 and aX^2 + bX + c with b^2 − 4ac
relationship between irreducibles and primes?
all primes are irreducible
in a UFD all irreducibles are also primes (except 0)
what is 1+1
2
subring test?
S is subset of R
closed unider +
closed under x
identity ∈ S
if F = ℱr R then R is a ?
subring of F
what does having an eisenstain prime mean?
f is irreducible
example of a UFD that is not principle?
K[X,Y] . as
any examples if domains that arn’t UFDs?
Z[√-5]
as 2x3 = (1 - √-5)(1 + √-5). 2 is irreducible in Z[√-5] but not prime
example of a noetherian domain that is not a PID?
K[X,X,…Xn] n>1
noetherian ring that is not a domain?
Zmod4 (or any mod, where the p is not prime)
a domain that is not noetherian?
Q[X,….Xn,…..]
R = domain
a maximal ideal is a … ?
prime ideal
