Spencer Leonardis4-14-2015

Homework 2Math 111B

Toolbox Math 111B

DefinitionsDefinition: Ring. A ring is a set R equipped with binary operations + : R×R → R and · : R×R → Rsatisfying the following axioms (note that we do not need to explictly state closure of addition andmultiplication, because binary operations defined this way are closed by definition):I. R is abelian group under addition:

1. a+b = b+a (+ is commutative)2. There exists 0 ∈ R such that 0+a = a+0= a for every a ∈ R. (Additive identity)3. For every a ∈ R there exists −a ∈ R such that a+ (−a)= (−a)+a = 0 (Existence of additive inverse)4. a+ (b+ c)= (a+b)+ c (+ is associative)

II. (Ring w/o identity). The multiplication of R, which we defined by · : R×R → R is associative.Or in other words, for every a,b, c ∈ R under · : R×R → R, one has a · (b · c)= (a ·b) · c.

II. (Ring with identity). R is a monoid under multiplication:1. (ab)c = a(bc) (· is a associative)2. There exists an element 1R ∈ R such that a ·1R = 1R ·a = a for every a ∈ R.

The multiplication of R, which we defined by · : R×R → R is associative. Or in other words, for everya,b, c ∈ R under · : R×R → R, one has a · (b · c)= (a ·b) · c.III. Multiplication distributes over addition

1. a(b+ c)= ab+ac for every a,b, c ∈ R.2. (b+ c)a = ba+ ca for every a,b, c ∈ R.

Definition: Commutative Ring. A ring R is called commutative if under the binary operation · : R×R → R,one has a ·b = b ·a for every a,b ∈ R.Definition: Division Ring. A ring R is said to be a division ring if its nonzero elements form a group undermultiplication (R \{0} forms a mult. group).Definition: Field. A commutative division ring.Definition: Zero-Divisor. Let R be a commutative ring. Then 0 6= a ∈ R is called a zero-divisor if there existsa b ∈ R,b 6= 0, such that ab = 0.Definition: Integral Domain. A commutative ring R is called an integral domain if ab = 0 implies thata = 0 or b = 0 for every a,b ∈ R. In other words, a commutative ring R is called an integral domain if itcontains no zero-divisors.Definition: Characteristic. Let R be a commutative ring. Then the characteristic of R, denoted Char(R) isthe smallest positive integer n such that na = a+·· ·+a︸ ︷︷ ︸

n summands

= 0 for every a ∈ R. If there exists a positive

integer n such that Char(R)= n then we say R has finite characteristic. If no such n exists, we say thatChar(R)= 0.Definition: Ring Homomorphism. Let R and S be rings. A mapping f : R → S is called a homomorphism if

f (a+b)= f (a)+ f (b) and f (ab)= f (a) f (b)

for every a,b ∈ R. We call a ring homomorphism f : a monomorphism if f is injective, an epimorphism if fis surjective. Isomorphism: f bijective.Definition: Kernel. The kernel of a ring homomorphism f , denoted ker( f ), is the set of all elements in Rthat get mapped to 0 (ie. the preimage f −1 ({0}) of the zero ideal.) Or more explicitly,

ker( f )= {a ∈ R : f (a)= 0}.

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Definition: Ideal. For a ring (R,+,◦) let (R,+) denote its additive group. A subset I is called a two-sidedideal or simply an ideal) of R if it is an additive subgroup of R that "absorbs multiplication by elements ofR." Formally we mean that I satisfies the following conditions:1. (I,+)É (R,+).2. For every x ∈ I and for every r ∈ R one has xr ∈ I (right ideal)3. For every x ∈ I and for every r ∈ R one has rx ∈ I (left ideal)Definition: Quotient ring. Let I�R. Then we define the quotient ring as R/I = {r+ I : r ∈ R} (ie the set ofcosets of I (abelian addition) in R). We define the addition and multiplication of R/I as(r+ I)+ (s+ I)= r+ s+ I and (r+ I)(s+ I)= rs+ I for every r, s ∈ R. We obtain that R/I is a ring under theseoperations, since the addition of R/I forms an abelian group and the multiplication of R/I is associative.The zero element of R/I is 0+ I = I.Definition: Princiapl Ideal Generated by a (ideal of all multiples of a.) Defined by

(a)= {ra : r ∈ R}= Ra.

Definition: Principal Ideal Domain. An integral domain R with unit is called a principal ideal domain ifevery ideal I is of the form I = (a)= Ra for some a ∈ R.Definition: Maximal Ideal. An ideal M 6= R in a ring R is called a maximal ideal of R if whenever U is anideal of R such that M ⊂U ⊂ R, then either R =U or M =U (ie. an Ideal M is maximal if we cannot fit anideal between M and the entire ring R).Definition: Euclidean Domain. An integral domain R is called a Euclidean domain if there exists a mapϕ : R \{0}→N such that1. For all nonzero a,b ∈ R, one has δ(ab)Ê δ(b).2. For every a,b ∈ R, b 6= 0, there exists q, r ∈ R such that a = bq+ r and r = 0 or ϕ(r)<ϕ(b).Definition: Unit. Let R be a ring with 1. An element u ∈ R is called a unit if there exists v ∈ R such thatuv = 1. Notation: U(R)= all units of R.Definition: Associates. Let R be a ring with 1. Two elements a,b ∈ R are called associates if b = ua forsome u ∈U(R).Definition: Prime. In a Euclidean domain R, a non-unit p is called prime if p = ab implies that a ∈U(R) orb ∈U(R).Definition: Relatively prime. In a Euclidean domain R, two elements are called relatively prime if theirgreatest common divisor is a unit in R.Definition: Degree of polynomial. If f (x)= a0 +a1x+·· ·+anxn and an 6= 0 then the degree of f (x) isdeg f (x)= n.Definition: Irreducible polynomial. A polynomial p(x) ∈ F[x] is called irreducible if wheneverp(x)= a(x)b(x) with a(x),b(x) ∈ F[x], then one of a(x) or b(x) has degree 0 (ie. either a(x) or b(x) areconstant (unit) polynomials).Definition: Content. The content of a polynomial 0 6= f (x)= a0 +·· ·+anxn ∈Z[x], denoted cont( f ) is thegreatest common divisor of a0, . . . ,an (ie the largest integer k such that f (x)/k still has integer coeficients).Definition: Primitive polynomial. A polynomial 0 6= f (x)= a0 +·· ·+anxn ∈Z[x], is called primitive ifcont( f )= 1.Definition: Monic polynomial. A polynomial 0 6= f (x)= a0 +·· ·+anxn ∈Z[x], is called monic if an = 1.

End definitions.

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Theorems/Lemmas/CorollariesLemma 3.2.1: Ring properties. If R is a ring, then for every a,b ∈ R:1. a0= 0a = 0.2. a(−b)= (−a)b =−(ab).3. (−a)(−b)= ab.If in addition, R has unit element 1, then:4. (−1)a =−a.5. (−1)(−1)= 1.Proof.1. Using distributive properties of ring.2. Additive identities and distributive properties.3. By part 2. and the fact that

(a−1)−1 = a (we use additive −(−a)= a in this case, but it’s the same) for

every a in a group.4. Additive identity and distributive property.5. Part 3.Lemma 3.2.2: A finite integral domain is a field.Corollary: If p is prime, then Z/pZ, the ring of integers modulo p, is a field.Lemma 3.3.1. Homomorphism properties: If ϕ is a ring homomorphism from R to S then1. ϕ(0)= 02. ϕ(−a)=−ϕ(a) for every a ∈ R.Lemma 3.3.2. Homomorphism properties: The kernel of a ring homomorphism ϕ : R toS satisfies:1. ker(ϕ) is an additive subgroup of R.2. If a ∈ ker(ϕ) and r ∈ R then ra,ar ∈ ker(ϕ).Lemma 3.3.3: A homomorphism ϕ : R → S is injective if and only if ker(ϕ)= {0} (ie ϕ is a monomorphism).Proof. Assume that ker(ϕ)= 0 and set ϕ(a)=ϕ(b). Then show that a+ (−b) ∈ ker(ϕ)⇐⇒ a+ (−b)= 0.Lemma 3.4.1: Let I�R and define π : R → R/I, r 7→ r+ I. Then π is an epimorphism (canonical projectionof R onto R/I) with ker(π)= I.Theorem. Ring Isomorphism Theorems: Let R and S be rings. If ϕ : R → S is a ring homomorphism, then:1. ker(ϕ)�R, Im(ϕ) is a subring of S and R/ker(ϕ)∼= Im(ϕ).2. x 7→ϕ(x) yields a bijection between subrings of R containing ker(ϕ) and subrings of Im(ϕ).

3. I →ϕ(I) yields a bijection between ideals of R. In this case, R/I ∼= Im(ϕ)ϕ(I)

.

4. If I, J�R then I + J�R andI + J

I∼= J

I ∩ J.

Lemma 3.5.1: Let R be a commutative ring with 1. Then R is a field if and only if its only ideals are (0)and R.Theorem 3.5.1: Let R be a commutative ring with 1 and let I�R. Then I is a maximal ideal if and only ifR/I is a field.Theorem 3.6.1: Every integral domain can be embedded into a field. That is, given a an integral domainD, one can construct a field of fractions F with a monomorphism ϕ : D → F.Lemma 3.6.1: Fix an integral domain D with S = {(a, r) : a, r ∈ D, r 6= 0}. Define a relation(a, r)∼ (b, s)⇐⇒ as = br. Then ∼ is an equivalence relation. We denote the equivalence class of (a, r) by a

r .The addition is defined as a

r + bs = as+br

rs and the multiplication is defined as ar

bs = ab

rs . The additive identityis 0= 0

r for every r 6= 0 and the multiplicative identity is 1= rr for every r 6= 0. We define the set of

equivalence classes (a, r) as F = { ar : a, r ∈ D, r 6= 0}.

Theorem 3.6.2: The set of equivalence classes Quot(D)= F = { ar : a, r ∈ D, r 6= 0} is a field.

Lemma 3.6.2: Let D be an integral domain and let F be its fraction field. Then the mappingφ : D → F, a → ar

r is a monomorphism. In particular, every integral domain D can be embedded in a field.Lemma 3.6.3: Let D denote an integral domain and F its fraction field. If K is a field and f : D → K is amonomorphism, then there exists a monomorphism g : F → K , g

( ar)= f (ar−1)= f (a) f (r)−1 whenever

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a, r ∈ D. The map g is a unique homomorphism such that g ◦φ= f , where φ denotes the embedding ofintegral domain D in fraction field F.Corollary: Let D be an integral domain with field of fractions F. If K is a field containing D, then Kcontains a subfield g(F)= Im(F)∼= F such that D ⊆ Im(F)⊆ K . (The image of fraction field F is mappedinto field K via a monomorphism)Proof. By the above Lemma, the inclusion mapping φ : D → F extends to a monomorphism g : F → K suchthat g(F)= Im(F)∼= F.Theorem 3.7.1: A Euclidean domain is a principal ideal domain.Corollary: Let D be a Euclidean domain with Ideal I. Then I = Da = {da : d ∈ D,0 6= a ∈ I} if and only ifδ(d)É δ(ad) for every 0 6= a ∈ I.Theorem: Every principal ideal domain contains unit element 1.Theorem: If R is a principal ideal domain and a,b ∈ R (not both 0) then gcd(a,b) exists and there arex, y ∈ R such that gcd(a,b)= xa+ yb.Corollary: If gcd(a,b)= 1 in R, then there exist x, y ∈ R if and only if 1= xa+ yb.Corollary: If R is a principal ideal domain where a | c, b | c, and gcd(a,b)= 1, then ab | c. If a | bc andgcd(a,b)= 1, then a | c.Lemma 3.7.2: Let R be an integral domain with 1 and suppose that for every a,b ∈ R, one has a | b andb | a. Then a = ub where u ∈U(R).Lemma 3.7.3: Let R be a Euclidean domain and let a,b ∈ R, a 6= 0. Then δ(ab)> δ(a) if b ∈U(R).Lemma: Let R be a principal ideal domain and let a, p ∈ R where p is prime. Then p - a if and only ifgcd(p,a)= 1.Corollary: If R is a principal ideal domain and p ∈ R is prime, then p | ab implies that p | a or p | b.Theorem: Any nonzero, nonunit element in a Euclidean domain can be expressed as a unique product ofprimes unique up to associate and permutation.Lemma: If p ∈ R is a prime, then up is prime for any u ∈U(R).Theorem 3.8.1: The integral domain of Gaussian integers, denoted Z[i], is a Euclidean domain.Lemma 3.8.1: Let p ∈Z be prime, and suppose that for some c ∈Z with (p, c)= 1, one has integers x, y ∈Zsuch that x2 + y2 = cp. Then p can be expressed as a sum two squared integers. In particular, there exista,b ∈Z such that p = a2 +b2.Lemma 3.8.2: If p is a prime of the form 4n+1, then we can solve the congruence x2 ≡−1 mod p.Theorem: Let R be a principal ideal domain. Then p ∈ R is prime if and only if (p)= is a maximal ideal.Corollary: If R is a principal ideal domain. Then p ∈ R is a prime if and only if the quotient ring R/(p), isa field. In particular, an Ideal is maximal in R if and only if R/I is a field.Lemma 3.9.1: If f (x) and g(x) are two nonzero elements in F[x], then deg( f (x)g(x))= deg f (x)+deg g(x).Corollary: If f (x) and g(x) are two nonzero elements in F[x], then deg f (x)É f (x)g(x).Theorem: The polynomial ring F[x] is a Euclidean domain with norm deg f (x) (and hence F[x] is a PIDand Integral domain) equipped with the following division algorithm: Given f (x), g(x) 6= 0 ∈ F[x] thereexist polynomials q(x), r(x) ∈ F[x] such that f (x)= g(x)q(x)+ r(x) with r(x)= 0 or deg r(x)< deg g(x).Lemma: Let 0 6= f (x) ∈ F[x]. Then a ∈ F is a root of f (x) if and only if f (x)= (x−a)g(x).Lemma: Let f (x) ∈ F[x] and 2É deg f (x)É 3. Then f (x) is reducible if and only if f (x) has a root a ∈ F.Lemma 3.9.4: Every polynomial in F[x] can be expressed uniquely as a product of irreduciblepolynomials.Lemma 3.9.5: Given two polynomials f (x), g(x) ∈ F[x], there exists a greatest common divisor d(x) ∈ F[x]such that d(x)= f (x)µ(x)+ g(x)λ(x) for some µ(x),λ(x) ∈ F[x].Lemma 3.9.6: The ideal (p(x)) ∈ F[x] is maximal if and only if p(x) is irreducible over F.Theorem:

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Theorem (One-step subgroup test). Let G be a group and let H 6= ; be a subset of G. Then H ÉG ifand only if for every a,b ∈ H one has ab−1.Proof.Theorem: kernel is subgroup. If f : G → H is a group homomorphism, then ker( f )ÉG.Proof. We know that ker( f ) is non-empty since f (1G)= 1H meaning that 1G ∈ ker( f ). Let a,b ∈ ker( f ) suchthat f (a)= f (b)= 1H . Then f (a−1b)= f (a−1) f (b)= f (a)−1 f (b)= 1H

−11H = 1H . Hencea−1b ∈ ker( f )⇒ ker( f )ÉG. 2

Let ϕ : R → S be a ring homomorphism:Theorem. ker(ϕ) is a subring of R (If we do not require that rings or subrings have a multiplicativeidentity).Proof. We know that (ker(ϕ),+)É (R,+) since the addition of a ring homomorphism is a grouphomomorphism and the kernel of a group homomorphism is a subgroup. Suppose that a,b ∈ ker(ϕ). Thenϕ(xy)=ϕ(x)ϕ(y)= 0S0S = 0S. Thus xy ∈ ker(ϕ).Theorem. ker(ϕ) is an ideal of R.Proof. We know that (ker(ϕ),+)É (R,+). Suppose a ∈ ker(ϕ) and r ∈ R. Then ϕ(ar)=ϕ(a)ϕ(r)= 0Sϕ(r)= 0S.Hence ar ∈ ker(ϕ) and similarly ra ∈ ker(ϕ). Thus ker(ϕ)�R. 2

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