Math 541 – 4/30 Apr 30, 2018 Shawn Math 541 No comments yet Math 541 – 4/27 Apr 30, 2018 Shawn Math 541 No comments yet Math 541 – 4/25 Apr 30, 2018 Shawn Math 541 No comments yet Math 541 – 4/23 Apr 29, 2018 Shawn Math 541 No comments yet 9.5 Equivalence Relations Apr 23, 2018 Shawn Math 240 No comments yet 1 … 13 14 15 16 17 … 60
![Ideals Generated by Subset • Definition ○ Let R be a commutative ring ○ If A is a subset of R, then the ideal generated by A is ○ (A)≔{r_1 a_1+…+r_n a_n│n∈Z(≥1),r_i∈R,a_i∈A}⊆R ○ If A is finite, then we write (A) as (a_1,…,a_n ) • Notice: when |A|=1, (A) is a principal ideal • Example: (2,x)⊆Z[x] ○ Suppose, by way of contradiction, that (2,x)=(p) for some p∈Z[x] ○ Since 2∈(p) § 2=pq for some q∈Z[x] § 0=deg2=degp+degq § degp=degq=0 ○ Since x∈(p) § Choose r∈R[x] s.t. pr=x, then deg〖r=1〗 § Write r=ax+b,where a,b∈Z § Then pr=p(ax+b)=x § So pa=1, by comparing coefficients § Since p∈Z[x] and a∈Z, p∈{±1} ○ Therefore (2,x)=Z[x] ○ So, 1=2p^′+xq′, where p^′,q^′∈Z[x] ○ Evaluating both side at 0, we get 1=2p^′ (0)=0 ○ This is a contradiction, so (2,x)⊆Z[x] • Example: Z[x]\/(2,x)≅Z\/(2) ○ Define F:Z[x]→Z\/2Z given by a_0 x^n+…+a_1 x+a_0↦(a_0 ) ̅ ○ F is a ring homomorphism § F factors as Z[x]→Z→Z\/2Z, where p↦p(0)↦p((0) ) ̅ § Composition of homomorphisms is still a homomorphism ○ F is certainly surjective ○ (2,x)⊆kerF § Let p∈(2,x) § Then p=2g+xh for some g,h∈Z[x] § Since xh has no constant term, and 2g has even constant term § F(p)=F(2g)=F(g)=0 ̅∈Z\/2Z ○ kerF⊆(2,x) § Let p∈kerF § Write p=a_n x^n+…+a_1 x+a_0 § Write a_0=2b,b∈Z § Then p=x(a_n x^(n−1)+…+a_1 )+2b∈(2,x) ○ Therefore, kerF=(2,x) ○ By the First Isomorphism Theorem of , Z[x]\/(2,x)≅Z\/2Z≅Z\/(2) ○ Note: Z[x]\/(x,n)≅Z\/(n) Maximal Ideal • An ideal M in a ring R is maximal if • M≠R, and the only ideals containing M are M and R Proposition 71: Criterion for Maximal Ideal • Statement ○ If R is a commutative ring, and M⊆R is an ideal ○ Then M is maximal ⟺R\/M is a field • Proof (⟹) ○ The only ideals containing M are R and M ○ Thus, R\/M has exactly 2 idals, by the Correspondence Theorem ○ Namely, the zero ideal, and the entire ring ○ Let x+M∈R\/M ○ Suppose x∉M i.e. x+M≠0 ○ Then (x+M)=R\/M ○ So 1+M∈(x+M) ○ Choose y+M∈R\/M s.t. (x+M)(y+M)=1+M ○ This shows x+M is a unit ○ Therefore R\/M is a field • Proof (⟸) ○ Suppose R\/M is a field ○ Then R\/M has exactly two ideals, 0 and R\/M ○ By the Correspondence Theorem, ○ There are exactly two ideals containing M: R and M ○ By definition of maximal ideal, M is maximal Examples of Maximal Ideals • What are the maximal ideals in Z? ○ (n)∈Z is maximal ⟺Z\/(n) is a field ⟺ n is prime • Is (x)⊆Z[x] maximal? ○ No, (x)⊊(2,x)≠Z[x] ○ Also, by First Isomorphism Theorem, Z[x]\/(x)≅Z, but Z is not a field § Define a ring map Z[x]→Z given by p→p(0) § F is surjective, and kerF=(x) • Is (x^2+1)⊆R[x] maximal? ○ R[x]\/(x^2+1)≅ℂ is a field • Is (x^2−1)⊆R[x] maximal ○ R[x]\/(x^2−1)≅R×R is not a field, since (1,0) is not a unit ○ Another way to see (x^2−1) is not maximal § (x^2−1)⊊(x−1)⊊R[x] § (x^2−1)⊊(x+1)⊊R[x]](https://shawnzhong.com/wp-content/uploads/2018/05/img_5aee4f89b32e7.png)
![Theorem 70: The First Isomorphism Theorem for Rings • Statement ○ If f:R→S is a ring homomorphism, then there is an induced isomorphism ○ f ̅:R\/kerf→im(f), given by r+kerf↦f(r) • Proof ○ We need only check f ̅(1_(R/kerf ) )=1_S, and f ̅ preserves multiplication ○ f ̅(1_(R/kerf ) )=f ̅(1+kerf )=f(1_R )=1_S ○ f ̅((r_1+I)(r_2+I))=f ̅(r_1 r_2+I)=f(r_1 r_2 )=f(r_1 )f(r_2 )=f ̅(r_1+I) f ̅(r_2+I) • Example: R[x]\/(x^2+1)≅ℂ ○ Let F:R[x]→ℂ given by p↦p(i) ○ F is a ring homomorphism § In fact, if R is a subgring of some ring S, and s∈S, then § The function R[x]→S given by p↦p(s) is a ring homomorphism ○ F is surjective § If a+bi∈ℂ, then F(a+bx)=a+bi ○ (x^2+1)⊆kerf § If p(x^2+1)∈(x^2+1), then § F(p(x^2+1))=F(p)F(x^2+1)=p(i)p(i^2+1)=0 ○ kerf⊆(x^2+1) § Let p∈kerf § Using polynomial division, we can find q,r∈R[x] s.t. § p=q(x^2+1)+r where degr deg(x^2+1)=2 § Write r=ax+b for some a,b∈R § Since p∈kerf, p(i)=0 § 0=p(i)=q(i)×(i^2+1)+r(i)=r(i)=ai+b § So a=b=0 § Therefore p=q(x^2+1), and p∈(x^2+1) ○ Therefore, kerf=(x^2+1) ○ By the First Isomorphism Theorem of Rings, R[x]\/(x^2+1)≅ℂ • Example: R[x]\/(x−a)≅R, where a∈R ○ Let F:R[x]→R given by p↦p(a) ○ F is surjective § F(b)=b,∀b∈R ○ F is a ring homomorphism ○ (x−a)⊆kerf § If p(x−a)∈(x−a), then § F(p(x−a))=F(p)F(x−a)=p(a)p(a−a)=0 ○ kerf⊆(x−a) § Let p∈kerf § Divide x−a into p to obtain q,r∈R[x] s.t. § p=q(x−a)+r, where degr 1 § Since p∈kerf, 0=p(a)=q(a)(a−a)+r=r § Thus r=0, so p=q(x−a)∈(x−a) ○ Therefore, kerf=(x−a) ○ By the First Isomorphism Theorem of Rings, R[x]\/(x−a)≅R • Example: R[x]\/(x^2−1)≅R×R ○ Recall: Chinese Remainder Theorem § If I,J are ideals in a commutative ring R s.t. I+J=R § Then R\/IJ≅R\/I×R\/J, where § I+J={x+y│x∈I,y∈J} § IJ={x_1 y_1+…+x_n y_n│n∈Z(≥1),x_i∈I,y_i∈J} ○ (x^2−1)⊆(x+1)(x−1) § This is obvious, since x^2−1∈(x+1)(x−1) ○ (x+1)(x−1)⊆(x^2−1) § Let p_1 q_1+…+p_n q_n∈(x−1)(x+1), where p_i∈(x−1),q_i∈(x+1) § Each term p_i q_i is of form □ f_i (x−1)⋅g_i (x+1)=f_i g_i (x^2−1) for some f_i,g_i∈R § Thus p_i q_i∈(x^2−1)⇒p_1 q_1+…+p_n q_n∈(x^2−1) ○ Thus (x^2−1)=(x+1)(x−1) ○ R[x]\/(x+1)(x−1)≅R×R § 1/2 (x+1)−1/2 (x−1)=1∈R[x] § ⇒(x+1)+(x−1)=R[x] § ⇒1∈(x+1)+(x−1) § Chinese Remainder Theorem implies R[x]\/(x+1)(x−1)≅R×R ○ Therefore, R[x]\/(x^2−1)≅R×R Other Isomorphism Theorems for Rings • The Second Isomorphism Theorem for Rings ○ If I is an ideal of a ring R, and S is a subring of R ○ Then S+I is also a subring of R, where ○ I is an ideal of S+I, and (S+I)\/I≅S\/(I∩S) • The Third Isomorphism Theorem for Rings ○ If I⊆J are ideals of a ring R, then (R\/I)⁄(J\/I)≅R\/J • Correspondence Theorem ○ If R is a ring, and I is an ideal of R ○ Then there is a bijection {ideals of R\/I}⟷{ideals of R containing I}](https://shawnzhong.com/wp-content/uploads/2018/05/img_5aec790d90b90.png)
![Polynomial Ring • Polynomials over a ring ○ Let R be a commutative ring ○ A polynomial over R is a sum a_n x^n+a_(n−1) x^(n−1)+…+a_1 x+a_0 ○ Where x is a variable, and a_i∈R • Degree ○ If f=a_n x^n+a_(n−1) x^(n−1)+…+a_1 x+a_0 is a polynomial over R ○ The degree of f, denoted deg(g), is sup{n≥0│a_n≠0} ○ Note: deg(0)=−∞ • Leading term and leading coefficient ○ If deg(f)=n≥0 ○ The leading term of f is a_n x^n ○ The leading coefficient of f is a_n • Notation ○ Let R[x]≔{Polynomials over a commutative ring R} ○ Then R[x] is a commutative ring with ○ ordinary addition and multiplication of polynomials • R is a subring of R[x] ○ R is identified with the constant polynomials ○ There is a ring homomorphism i:R→R[x] defined as ○ mapping the ring element r∈R to the constant polynomial r ○ The constant polynomials in R[x] form a subring ○ And i gives an isomorphism between R and the subring • Polynomial ring with multiple variables ○ We define polynomial rings in several variables inductively ○ R[x_1,x_2 ]=(R[x_1 ])[x_2 ] ○ ⋮ ○ R[x_1,…,x_n ]=(R[x_1,…,x_(n−1) ])[x_n ] Proposition 67: Properties of Polynomial Rings • Statement ○ Let R be a domain. Let p,q∈R[x]∖{0}, then 1. deg(pq)=deg(p)+deg(q) 2. (R[x])^×=R^× 3. R[x] is a domain • Proof ○ Write § p=a_n x^n+…+a_1 x+a_0, where deg(p)=n § q=b_m x^m+…+b_1 x+b_0, where deg(q)=m ○ Then a_n≠0 and b_m≠0 ○ Since R is a domain, a_n m_m≠0 ○ So, the leading term of pq is a_n b_m x^(m+n), which verifies (1) ○ Also, a_n b_m x^(m+n)≠0. This proves (3) ○ For (2), suppose pq=1, then § deg(p)+deg(q)=deg(pq)=0 by (1) § Thus, deg(p)=0=deg(q) i.e. p,q∈R § Since pq=1, p,q∈R^× § Thus (R[x])^×⊆R^× § Also, R^×⊆(R[x])^× § Therefore (R[x])^×=R^× Ideal • Definition ○ Let R be a ring, let I be a subset of R, and let r∈R ○ rI≔{rx│x∈I} is a left ideal of R if § I is a additive subgroup of R § rI=I, ∀r∈R ○ Right ideal Ir≔{xr│x∈I} are defined similarly ○ I is an ideal if I is both a left and right ideal • Intuition ○ Normal subgroups are to groups as ideals are to rings • Example ○ If R is a ring, then R and {0} are both ideals Proposition 68 • Statement ○ If I⊆R is an ideal, then I=R⟺1∈I • Proof ○ (⟹) Trivial ○ (⟸) Let r∈R. ○ By definition of ideal, rI=I ○ So r=r⋅1∈I ○ Thus R=I • In particular, if R is a ring, and S is a subring of R ○ S is an ideal in R⟺S=R (subrings contain 1) ○ Similarly, if I⊆R is an ideal, and I is a subring of R⟺I=R Principal Ideal • Definition ○ Let R is a commutative ring, and let r∈R, then ○ (r)≔{ar│a∈R} is called the principal ideal generated by r • Proof: principal ideals are ideals ○ 0=0⋅r∈(r), so (r) is not empty ○ Let ar,br∈(r), then § ar−br=(a−b)r∈(r) § Therefore, (r) is an additive subgroup of R ○ Let a∈R, br∈(r), then § a(br)=abr∈(r) § (br)a=bra=abr∈(r) § So a(r)=(r)a,∀a∈R • Example ○ If n∈Z, (n) is just the cyclic subgroup generated by n](https://shawnzhong.com/wp-content/uploads/2018/05/img_5aec78da6ebc7.png)
![Equivalence Relations • Definition 1 ○ A relation on a set A is called an equivalence relation if ○ it is reflexive, symmetric, and transitive • Definition 2 ○ Two elements a,b that are related by an equivalence relation are called equivalent ○ The notation a ∼ b is often used to denote that a and b are equivalent elements with respect to a particular equivalence relation. Strings • Example ○ Suppose that R is the relation on the set of strings of English letters such that ○ aRb if and only if l(a) = l(b), where l(x) is the length of the string x. ○ Is R an equivalence relation? • Solution ○ Show that all of the properties of an equivalence relation hold ○ Reflexivity § Because l(a)=l(a), it follows that aRa for all strings a. ○ Symmetry § Suppose that aRb. Since l(a)=l(b), l(b)=l(a) also holds and bRa. ○ Transitivity § Suppose that aRb and bRc § Since l(a)=l(b),and l(b)=l(c), l(a)=l(a) also holds and aRc. Congruence Modulo m • Example ○ Let m be an integer with m 1 ○ Show that the relation ○ R={(a,b)│a≡b (mod m) } ○ is an equivalence relation on the set of integers. • Solution ○ Recall that a≡b (mod m) if and only if m divides a−b ○ Reflexivity § a≡a (mod m) since a−a=0 is divisible by m since 0 = 0 ∙ m. ○ Symmetry § Suppose that a≡b (mod m) § Then a−b is divisible by m, and so a−b=km, where k is an integer § It follows that b−a=(−k)m, so b≡a (mod m). ○ Transitivity § Suppose that a≡b (mod m) and b≡c (mod m). § Then m divides both a−b and b−c. § Hence, there are integers k and l with a−b=km and b−c=lm § We obtain by adding the equations: § a−c=(a−b)+(b−c)=km+lm=(k+l)m § Therefore, a≡c (mod m) Divides • Example ○ Show that the “divides” relation on the set of positive integers is not an equivalence relation. • Solution ○ The properties of reflexivity, and transitivity do hold, but there relation is not transitive. ○ Hence, “divides” is not an equivalence relation. ○ Reflexivity § a ∣ a for all a. ○ Not Symmetric § For example, 2 ∣ 4, but 4 ∤ 2 § Hence, the relation is not symmetric. ○ Transitivity § Suppose that a divides b and b divides c. § Then there are positive integers k and l such that b=ak and c=bl. § Hence, c=a(kl), so a divides c. § Therefore, the relation is transitive. Equivalence Classes • Let R be an equivalence relation on a set A. • The set of all elements that are related to an element a of A is called the equivalence class of a • The equivalence class of a with respect to R is denoted by [a]_R. • When only one relation is under consideration, we can write [a], without the subscript R • Note that [a]_R={s|(a,s)∈R} • If b∈[a]_R, then b is called a representative of this equivalence class. • Any element of a class can be used as a representative of the class. • The equivalence classes of the relation congruence modulo m are called the congruence classes modulo m. • The congruence class of an integer a modulo m is denoted by [a]_m • So [a]_m={…,a−2m,a−m,a+2m,a+2m,…} • For example, ○ [0]_4 = {…, −8, −4 , 0, 4 , 8 , …} ○ [1]_4 = {…, −7, −3 , 1, 5 , 9 , …} ○ [2]_4 = {…, −6, −2 , 2, 6 , 10 , …} ○ [3]_4 = {…, −5, −1 , 3, 7 , 11 , …} Equivalence Classes and Partitions • Theorem 1 ○ Let R be an equivalence relation on a set A. ○ These statements for elements a and b of A are equivalent: i) aRb ii) [a]=[b] iii) [a]∩[b]=∅ • Proof ○ We show that (i) implies (ii). ○ Assume that aRb. ○ Now suppose that c ∈ [a].Then aRc. Because aRb and R is symmetric, bRa. ○ Because R is transitive and bRa and aRc, it follows that bRc. ○ Hence, c ∈ [b]. Therefore, [a]⊆ [b]. ○ A similar argument (omitted here) shows that [b]⊆ [a]. ○ Since [a]⊆ [b] and [b]⊆ [a], we have shown that [a] = [b]. Partition of a Set • A partition of a set S is a collection of disjoint nonempty subsets of S that have S as their union. • In other words, the collection of subsets A_i, where i∈I, forms a partition of S if and only if ○ A_i≠∅ for i∈I, ○ A_i∩A_j=∅ when i≠j, ○ ⋃8_(i∈I)▒A_i =S An Equivalence Relation Partitions a Set • Let R be an equivalence relation on a set A. • The union of all the equivalence classes of R is all of A • Since an element a of A is in its own equivalence class [a]_R. In other words, ○ ⋃8_(a∈A)▒[a]_R =A • From Theorem 1, it follows that these equivalence classes are either equal or disjoint • So [a]_R∩[b]_R=∅ when [a]_R≠[b]_R. • Therefore, the equivalence classes form a partition of A • Because they split A into disjoint subsets. Equivalence Relation and Partition • Theorem 2 ○ Let R be an equivalence relation on a set S. ○ Then the equivalence classes of R form a partition of S. ○ Conversely, given a partition {A_i│i∈I} of the set S ○ There is an equivalence relation R that has the sets A_i, i∈I, as its equivalence classes. • Proof ○ We have already shown the first part of the theorem. ○ For the second part, assume that {A_i│i∈I} is a partition of S. ○ Let R be the relation on S consisting of the pairs (x, y) ○ where x and y belong to the same subset A_i in the partition. ○ We must show that R satisfies the properties of an equivalence relation. ○ Reflexivity § For every a∈S, (a,a)∈R, because a is in the same subset as itself. ○ Symmetry § If (a,b)∈R, then b and a are in the same subset of the partition, so (b,a)∈R ○ Transitivity § If (a,b)∈R and (b,c)∈R, then a and b are in the same subset of the partition, as are b and c. § Since the subsets are disjoint and b belongs to both, the two subsets of the partition must be identical. § Therefore, (a,c)∈R since a and c belong to the same subset of the partition.](https://shawnzhong.com/wp-content/uploads/2018/04/img_5addedc1250d8.png)
