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Sin título
Fichas del conjunto (36)
- 1What's a ring?- A set of elements R
- Two commutative, associative, binary operations. +, ×
- Mapping φ: R×R→R
- Contains 1ᵣ - 2When is a ring an abelian group?- under addition
- with a unique element 0ᵣ∈R - 3What distributive laws does a ring follow?a(b+c) = ab+ac
- 4What is an ideal?- I⊲R
- So I is a subset of R
- 0∈I
- for a,b∈R then a-b∈I
- for all a∈I and r∈R ar∈I - 5What's the quotient ring R/I?Gives the structure of the cosets such that:
- (r₁+I) + (r₂+I) = (r₁+ r₂) + I
- (r₁+I)(r₂+I)= r₁ r₂ +I
for all r₁, r₂∈R
The zero element of R/I is Oᵣ + I=I
and similarly 1ᵣ+I - 6What is a proper ideal?I ≠ R
- 7What's a maximal ideal?M⊲R
For M⊂I⊂R then either I=M or I=R
(There are no ideals between M and R) - 8What is a prime ideal?- A proper ideal
- P⊲R
For (ab)∈P for a,b∈R then either a or b (or both) ∈P - 9What is a principal ideal?I⊲R if ∃a∈R such that I={ra : r∈R}
the principal ideal generated by a∈R is denoted by (a) - 10Name two principal ideals?1) The trivial ideal {0}=(0)
2) The whole ring R=(1) - 11What is a subring?A subset A of R with:
- 1ᵣ ∈ A
- for a,b∈R a+b, ab∈A
- for a∈R then a⎺ ¹∈ A - 12What is a ring homomorphism?φ: R→S
- φ(r₁+r₂)=φ(r₁)+φ(r₂)
- φ(r₁r₂) = φ(r₁)φ(r₂)
- φ(1ᵣ)=1푠 - 13What is the Fundamental Theorem of Ring Homomorphisms?φ: R→S is a ring homomorphism if:
Im(φ) ≃ R/ Ker(φ)
Where Im(φ) is a subring of S and Ker(φ) is an ideal of R - 14What is a field?A ring in which every non-zero element has a multiplicative inverse
whenever a≠0 ∈ F ∃b∈F such that ab=1 - 15What are the only ideals of a field?1) The trivial ideal {0}
2) The field itself - 16When is R/M a field?IFF
M is maximal for M⊲R - 17What is an integral domain?A ring R with no zero-divisors
So for a,b∈R if ab=0 then a=0 or b=0 - 18In an integral domain the trivial ideal {0} is ...prime
- 19When is R/P an integral domain?IFF
P⊲R and P is a prime ideal - 20Maximal ideals are ...prime
- 21What is a unit?u∈R is a unit if it has a multiplicative inverse u⎺ ¹ in R
- 22What is the ideal generated by a unit?(u)=R
- 23DividesGiven a,b∈R then a | b iff ad=b for some d∈N
- 24What is an irreducible element?q∈R is irreducible if it is non-zero and non-unit and whenever q=ab for some a.b∈R then either a is a unit or b is a unit
- 25What is a prime element?For p∈R then p┃ab implies that p┃a or p┃b (or both)
- 26In an integral domain any prime element is ....Irreducible
- 27For p∈ R\(0) then p is prime iff ....(p) is a prime ideal
- 28What is a polynomial ring?R[x] is a ring R in which all the coefficients of x ∈ℤ
- 29If R is an integral domain then is R[x]?Yes
- 30For ring R and R[x] then for f,g ≠0 then deg(fg)= ......?deg (fg) = deg (f) + deg (g)
- 31What is a principal ideal domain?An integral domain in which every ideal if principal
- 32Give 2 examples of PID1) ℤ
2) F[x] - 33If R is a PID and q∈R is irreducible then what is (q)?- a maximal ideal
- q is prime - 34What is an Unique Factorization Domain?An integral domain with
1) For every nonzero nonunit a∈R there is a factorization a=q₁ q₂ .... q푛
with each q ᵢ irreducible
2) Given a=q₁ q₂ ... q푛= r₁ r₂ .... r횖 then 푛=횖 and ∃u₁, ... u푛∈R such that q ᵢ = uᵢ rᵢ for all ᵢ - 35All PIDs are ...UFDs
- 36In a UFD all irreducibles are ...prime
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- Fichas sobre Sin título, creado por Tara Annison el 17/02/2015.
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