Ongoing deck, Billy Cole Flashcards
what is the ring Z_n
the ring of integrers modulo , with addition and multiplication modulo n.
define a Ring. and its 4 conditions.
a ring is given by the following: a set R and two binary operations, +, x, on R. Conditions are;
R1. (R,+) is an abelian group with e=0
R2. x is associative
R3. x is distributive over +
R4 there exists an identity for x, that is not 0
lemma 1.3. state the Ring rules.
can you prove each one?
i, 0a=a0=0
ii, a(-b) = (-a)b = -(ab)
iii, (-a)(-b) = ab
Lemma 1.5. If S⊆R then S is a subring of R iff ?
1∈ S
r,s ∈ S ⇒ r+s, rxs ∈ S
r ∈ S ⇒ -r ∈ S
define a subring.
R = a ring and S⊆R. S is a subring of R if it is a ring in its own right with respect to the same addition and multiplication as in R.
definition 1.8. define Polynomial Rings.
let R be a ring. the ring of polynomials R[X] in the indeterminate X is defined as follows: Elements. ∑ ( a_iX^i ) Equality Addition Multiplication
what is the degree of a polynomial?
for ∑ ( a_iX^i ), we define,
deg(f) = the largest i such that A_i ≠0
what is the deg(f=0) ?
minus infinity
formula for deg(fg) ?
deg(fg) ≤ deg(f) + deg(g)
formula for deg(f+g) ?
deg(f+g) ≤ max(deg(f),deg(g))
def 2.1. define the characteristic, char(R)
the characteristic of a ring R is the least positive integer n such that 1+…+1 =0. that is such that n∙ 1=0.
lemma 2.2. suppose that char(R) = n>0 then ?
can you prove the two?
i) n∙ r=0 for every r∈R
ii) if m is a positive integer then m∙ 1=0. iff n divides m.
definition2.3. define the domain.
the ring R is a domain if, for all r,s ∈ R
rs=0 ⇒ r=0 or s=0.
define a zero divisor
a non-zero element r∈R is a zero divisor if there is a non-zero element s∈R with rs=0 or sr=0. so a domain is a ring without any zero divisors.
what is a integral domain?
an integral domain is a commutative domain.
lemma 2.5. S⊆R and R is a domain. then ?
S is also a domain, can you prove this? easy
♕
♛
propostion 2.7.
if R is a domain, then what is R[X]?
also a domain. can you prove this?
corollary 2.8.
R is a domain. then the ring R[x_1,…,X_n] is what?
also a domain, as long as R[x_1,…,X_n] is a ring of polynomials in n in-determinates and with coefficients in R.
definition 2.9.
a division ring ?
a division ring is a ring in which every non zero element has a right inverse and a left inverse: for every r∈R there is s∈R s.t. rs=1. and there is t∈R s.t. tr=1.
define a field
a field is a commutative division ring
Lemma 2.11.
R=ring and r∈R has both right and left inverse then ??
these are equal
can you prove ?
lemma 2.12.
what are the following conditions on the interger n≥2 that are equivalent.
- ℤ_n is an integral domain
- ℤ_n is a field
- n is a prime
Proposition 2.14.
every ?? is a ??. and every ?? is an ??
every division ring is a domain.
and every field is an integral domain
lemma 2.16
in any ring R. the set of units (inverses) R* forms a ???
in any ring R. the set of units (inverses) R* forms a group under multiplication
definition 2.17
define nilpotent
an element r∈R is nilpotent if there is some integer n≥1 with r^n =0.
define the index of nilpotence
the least such n that r^n =0.
define idempotent
an element r∈R is idempotent if r²=r
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definition 3.1
define isomorphism
if R&S are rings then an isomorphism from R to S is a bijection θ:R↦S such that, for all r,r’∈R we have
θ(r + r’) = θ(r) + θ(r’)
θ(r x r’) = θ(r) x θ(r’)
the left operations are in R and the right operations are in S
define R and S are isomorphic
R and S are isomorphic, is there is an isomorphism from R to S
lemma 3.3
suppose that θ:R↦S is an isomorphism. then ….?
- θ(1)=1
- θ(0)=0
- θ(-r) = -θ(r) for every r in R
- r∈R is invertible iff θ(r)∈S is invertible and, in that case, (θ(r))^-1 = θ(r^-1)
- r∈R is nilpotent iff θ(r) is nilpotent. (they then have the same index of nilpotence )
definition 3.6
define homomorphism
If R and S are rings then a homomorphism from R to S is a map θ:R↦S such that, for all r,r’∈R we have,
θ( r+r’ ) = θ(r) + θ(r’)
θ( rxr’ ) = θ(r) x θ(r’)
we also require that θ(1_R)=1_S
lemma 3.7
suppose that θ:R↦S is an homomorphism. then ….?
- θ(0)=0
- θ(-r) = -θ(r) for every r in R
- if r∈R is invertible then θ(r)∈S is invertible and, in that case, (θ(r))^-1 = θ(r^-1)
- if r∈R is nilpotent then θ(r) is nilpotent.
- the image of θ is a subring of S.
what is an monomorphism (or an embedding)
it ius an injective (one to one) homomorphism
lemma 3.10a
if θ:R↦S and β:S↦T are homomorphisms then ???
so is the composition βθ:R↦T.
lemma 3.10b
if θ:R↦S and β:S↦T are embeddings then ??
so is the composition βθ:R↦T.
lemma 3.10c
if θ:R↦S and β:S↦T are homomorphisms and if βθ:R↦T is an embedding then ???
θ is an embedding
lemma 3.12
if θ:R↦S is a homomorphism then θ is ???
injective iff ker(θ) = {0}
definition 3.11
define the kernel
if θ:R↦S is a homomorphism of rings then the kernel of θ, ker(θ), is the set {r∈R | θ(r)=0}, of elements which θ sends to 0_S.
define an automorphism.
an automorphism of a ring is an isomorphism from the ring to itself.
ie the identity map is always an automorphism.
lemma 3.17a
suppose that θ:R↦S is a homomorphism. then the ker(θ) is ???
a subgroup of (R,+)
lemma 3.17b
let r,r’∈R. then ??? iff ??? iff ????
let r,r’∈R. then θ(r)=θ(r’) iff r-r’ ∈ker(θ) iff r and r’ belong to the same coset of ker(θ) in R
definition 3.18
define an ideal
an ideal of a ring R is subset I⊆R such that:
0∈I
a,b ∈I ⇒ a+b∈I
a∈I and r∈R ⇒ ar∈I and ra∈I
definition 3.19
define the principle ideal generated by
cant be bothered to write this out
Define a principle ideal
An ideal that can be generated by a single element
define the trivial ideal
in every ring = {0} is the smallest ideal.
define the proper ideal.
in every ring =R is the largest ideal and every other ideal is referred to as a proper ideal
define a right ideal
same definition as an ideal but with a weaker third condition;
a∈I and r∈R ⇒ ar∈I.
left ideals are defined similarly
define the principle right ideal generated by
a∈R is defined to be the set {ar | r∈R} and is denoted aR
proposition 3.22
a commutative ring R is a field iff ?
the only ideals of R are {0} and R
proposition 3.24
if θ:R↦S is a homomorphism of rings then ker(θ) ???
is an ideal of R
corollary 3.25
if θ:R↦S is a homomorphism of rings and R is a field then ???
then θ is a monomorphism
proposition 3.26
suppose that I and J are ideals of the ring R, then ??
- I + J ={a+b | a∈I, b∈J} is an ideal
- I⋂J is an ideal
- if {I_λ}_λ is any collection of ideals of R then their intersection ∩_λI_λ is an ideal
Factor Rings
U WOT MATE !?!
Define the set of cosets of I in the additive group .
Is this definition well defined?
R/I = {r + I | r∈R}
under operations;
( r + I ) + ( s + I) = ( r + s ) + I
( r + I ) x ( s + I) = ( r x s ) + I
Lemma 4.2- the operations are well defined, can you prove it ?Lemma 4.3. Let R be a ring and let I be a proper ideal. the set R/I is a ring.
what is the the zero element?
what is the identity element?
what is the inverse ? (provided r has an inverse)
The zero element is the coset 0+I (=I)
The identity element is 1+I
- ( r + I ) = ( -r ) + I and if r has an inverse in R then ( r + I ) is invertible in R/I with inverse r^-1 + I.
Define the Factor Ring of R by I.
The set of cosets R/I = {r + I | r∈R}.
under operations;
( r + I ) + ( s + I) = ( r + s ) + I
( r + I ) x ( s + I) = ( r x s ) + I
with
The zero element is the coset 0+I (=I)
The identity element is 1+I
- ( r + I ) = ( -r ) + I and if r has an inverse in R then ( r + I ) is invertible in R/I with inverse r^-1 + I.State the Fundamental Isomorphism Theorem
I = proper ideal of ring R
(i) the map π: R↦R/I, defined by π(r) = r + I, is a surjective ring homomorphism with kernal I.
(ii) if θ:R ↦ S is a homomorphism and I⊆ker(θ) then there is a unique map θ’ : R/I ↦ S with θ’◦π = θ. θ’ is a homom
(iii) the map θ’ is injective iff ker(θ) = I. If θ is surjective and ker(θ) = I then θ’ is an isomorphism
thm 4.12. if I≤J are ideals of R, so J/I is an ideal of R/I, then??
(R/I)/(J/I) ≃ R/J
thm 4.8. let I be an ideal of the ring R. Then there is a natural, inclusion-preserving bijection between the set of ????
of ideals of R which contain I and the set of ideals of the factor ring R/I.
Define Maximal
an ideal I of a ring is maximal if it is proper and of there is no ideal between it and R.
if any ideal J with I≤J≤R, then J=I or J=R.
corollary 4.11. if R is a commutative ring then an ideal I◅R is maximal iff ??
the factor ring R/I is a field.
Define a prime proper ideal
a proper ideal I of a commutative ring R is prime whenever r,s ∈R and rs ∈I then either r ∈I or s ∈I
POLYNOMIAL RINGS AND FACTORISATION
⚢ + ♂ = ?
GREAT FUN
thm 5.1. State the division theorem for polynomials.
what are the quotient and remainder
let K be a field and take f,g ∈K[X] with g≠0. then there are unique q,r ∈K[X] with
f=qg + r and deg(r) < deg(g) or r = 0.
q is the quotient and r is the remainder when f is divided by g.
Define a Root (of f)
An element a ∈ K is a root ( or a zero) of f ∈K[X] if f(a) = 0
Corollary 5.4. ??
roots
K is a field. f ∈K[X] and a ∈K. then a is a root of f iff X - a is a factor of f.
Define the Greatest common divisor (or HCF) of polynomials.
is a polynomial d such that d divides f and g and if h is any polynomials dividing both f and g then h divides d. write d=gcd(f,g). this polynomial is defined only up to a non-zero scalar multiple so, if we want a unique gcd then we can insist that d be monic.
Describe the Division algorithm
-
Corollary 5.6.
gcd
Let K be a field and take f,g ∈ K[X]. Then the ideal
generated by f and g equals the ideal generated by their greatest common divisor:
Corollary 5.8.
Let K be a field. Then every ideal of K[X] is principal (i.e. can be generated by a single polynomial).
Define Irreducible
An element r ∈ R is irreducible if r is not invertible and
if, whenever r = st either s or t is invertible.
Define Associated
Elements r; s ∈ R are associated if s = ur for some invertible element u ∈ R: For instance two integers r; s are associated iff r = ±s.
Define a Unique Factorisation Domain
A commutative domain R is said to be a unique factorisation domain if every non-zero, non-invertible element of R has an essentially unique factorisation as a product of irreducible elements.
Define a Principal Ideal Domain
A principal ideal domain, is a commutative
domain in which every ideal is principal (that is, is generated by some single element).
Lemma 6.2.
Let K be a field and let f ∈K[X] be irreducible. Then..
is a maximal ideal of K[X].
Thm 6.3. State Kronecker’s Theorem
Let K be a field and let f ∈K[X] be irreducible of degree n. Define L = K[X]/ then;
(i) L is a field and homomorphism π: K[X] ↦ K[X]/ induces an embedding θ: K↦L.
(ii) α= π(X)∈L is a root of f
(iii) The dimension of L as a vector space over K is n, with {1,α,α^2,…, α^n-1 } being a basis of L over K, so every element of L has a unique representation of the form a_n-1α^n-1 + … + a_1α + a_0 with a_n-1,…a_0 ∈K
Define Algebraically Closed
A field K is said to be algebraically closed if any polynomial f ∈K[X] has all its root in K.
