Ideals, congruence modulo ideal, factor rings

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1 Ideals, congruence modulo ideal, factor rings Sergei Silvestrov Spring term 2011, Lecture 6 Contents of the lecture Homomorphisms of rings Ideals Factor rings Typeset by FoilTEX

2 Congruence in F[x] and congruence classes This short repetition from previous Lecture 5 is important example for material on ideals and factor rings, which is the main topic of this Lecture 6 F is a field. p(x) F[x] is the ring of polynomials. Definition 1. (Sec 5.1, p. 119) f g(mod p) are congruent modulo p if and only if p(x) ( f (x) g(x)) Theorem 1. (Theorem 5.1, p. 120) Congruence is equivalence relation on F[x]. Definition 2. Congruence classes are defined as equivalence classes corresponding to the equivalence relation [ f ] = {g: g f (mod p)} = { f (x) + k(x)p(x): k(x) F[x]} Theorem 2. (Th. 5.3, p. 121) f (x) g(x)(mod p(x)) [ f ] = [g] OBS! [0] = {g: g 0(mod p)} = {k(x)p(x): k(x) F[x]} is closed under multiplication by any h(x) F[x] since h(x)(k(x)p(x)) = (h(x)k(x))p(x) In the general terminology we will study in this lecture, [0] is an ideal in F[x] and [ f ] is an element of a factor rings of the ring F[x] by the ideal [0]. Typeset by FoilTEX 1

3 Ring homomorphisms This repetition from previous Lectures is important for material on ideals and factor rings, which is the main topic of this Lecture 6 Definition 3. If R and R are rings, a ring homomorphism is a function ϕ : R R such that 1. ϕ(a + b) = ϕ(a) + ϕ(b) for all a, b R. 2. ϕ(ab) = ϕ(a)ϕ(b) for all a, b R. Example 1. Let R be an integral domain, and let F be its field of quotients. The function ϕ : R F given by ϕ(a) = [a,1] is easily seen to be a homomorphism. Example 2. Let R be a ring, and let R[x] be its ring of polynomials. The function ϕ : R R[x], given by ϕ(a) = (a,0,...) = a is a homomorphism. Example 3. Complex conjugation z = a + bi z = a bi is a homomorphism C C. Example 4. Choose m 2 and define the ring homomorphism ϕ : Z Z m by f (n) = n mod m, that is f (n) = [n] Z m is the congruence class of n modulo m. Typeset by FoilTEX 2

4 Properties of homomorphisms Theorem 3. If ϕ : R R is a ring homomorphism, then, for all a R, 1. If R has unity 1, then ϕ(1) is unity for ϕ[r]. 2. ϕ(a n ) = ϕ(a) n for all n If a is a unit, then ϕ(a) is a unit and ϕ(a n ) = ϕ(a) n for all n 1. Proof. 1. For all a R, ϕ(a) = ϕ(1a) = ϕ(a1) = ϕ(1)ϕ(a) = ϕ(a)ϕ(1), so ϕ(1) is unity for ϕ[r]. 2. Induction on n If ab = 1, then 1 = f (ab) = f (a) f (b), so ϕ(a 1 ) = ϕ(a) 1. Then use induction on n 1. Typeset by FoilTEX 3

5 Kernels and ideals Definition 4. The kernel of a homomorphism of rings ϕ : R R is its kernel as a map of additive groups; that is, Ker(ϕ) = ϕ 1 (0). Definition 5. (sec 6.1, p. 135) A subset I of a ring R is an ideal (or two-sided ideal when ring R is non-commutative) if 1. I is an additive subgroup of R, which means that it is a subset of R closed under addition in R; 2. if r R and a I, then ar I and ra I. The equivalent reformulation of this defintion is Definition 6. (sec 6.1, p. 135) A subring I of a ring R is an ideal (or two-sided ideal when ring R is non-commutative) if r R, a I ar I, ra I. Example 5. Two ideals of a ring R are R itself (improper ideal) and {0} (trivial ideal). Example 6. For each integer n the cyclic subgroup nz is an ideal in Z. Example 7. For any subset S R the set of real or complex valued (continuous) functions vanishing on S (that is f (x) = 0 for all x S) is an ideal in the ring of all (continuous) functions C(R). Typeset by FoilTEX 4

6 A parallel with group theory Glimpse into the future lectures on groups Ideals play approximately the same role in the theory of rings as normal subgroups do in the theory of groups. For instance, let R be a ring and I an ideal of R. Since the additive group of R is abelian, I is a normal subgroup. Consequently, there is a welldefined factor group R/I in which addition is given by (a+i)+(b+i) = (a+b)+i. R/I can in fact be made into a ring. As one might suspect from the analogy with groups, ideals and homomorphisms of rings are closely related. Various isomorphism theorems for groups carry over to rings with normal subgroups and groups replaced by ideals and rings respectively. In each case the desired isomorphism is known to exist for additive abelian groups. If the groups involved are, in fact, rings and the normal subgroups ideals, then one need only verify that the known isomorphism of groups is also a homomorphism and hence an isomorphism of rings. Typeset by FoilTEX 5

7 Factor rings from homomorphisms Theorem 4. (Th. 6.10, Sec 6.2, p 147) The kernel of a ring homomorphism ϕ : R R from a ring R to a ring R is an ideal in R. Ker(ϕ) = {r R: ϕ(r) = 0 R } = ϕ 1 (0 R ) Theorem 5. (Theorem 6.9, Sec 6.2, p 147) Let R be a ring and I an ideal of R. Then the additive factor group R/I is a ring (factor ring) with multiplication given by (a + I)(b + I) = ab + I. If R is commutative or has a unity, then the same is true of R/I. Proof. Once we have shown that multiplication in R/I is well defined, the proof that R/I is a ring is routine. Suppose a + I = a + I and b + I = b + I. We must show that ab + I = a b + I. Since a a + I = a + I, a = a + i for some i I. Similarly, b = b + j with j I. Consequently a b = (a + i)(b + j) = ab + ib + a j + i j. Since I is an ideal, a b ab = ab + a j + i j I. Therefore a b + I = ab + I, whence multiplication in R/I is well defined. Example: the residue classes Example 8 (Example revisited). Let R = Z, let R = Z n and let γ : Z Z n maps an integer m Z to the reminder γ(m) when m is divided by n. γ is a homomorphism of rings. The kernel of γ is nz. By Theorem 6, the factor ring Z/nZ is isomorphic to Z n. The cosets of nz are the residue classes modulo n. The isomorphism γ : Z/nZ Z n assigns to each residue class its smallest nonnegative element. Theorem 6. (Theorem 6.12, Sec 6.2, p 148) Let R be a ring and I an ideal of R. Then the map π : R R/I given by is a surjective homomorphism, and its kernel is π(r) = r + I Ker(π) = I Typeset by FoilTEX 6

8 Algebra course FMA190/FMA190F Factor rings from ideals Theorem 7. Let I be an additive subgroup of a ring R. The coset multiplication (a + I)(b + I) = (ab) + I is well defined, independent of the choices a and b from the cosets, and makes the group R/I of left cosets into a ring if and only if I is an ideal of R. The first isomorphism theorem for rings Theorem 8. (The first isomorphism theorem for rings) (Th. 6.13, Sec. 6.2, p 149) If ϕ : R R is a homomorphism with kernel K, then ϕ[r] is a ring, and µ : R/K Im(ϕ) R given by µ(a + K) = ϕ(a) is an isomorphism. If γ : R R/K is the homomorphism given by γ(a) = a + K, then ϕ = µ γ. R γ ϕ R/K ϕ[r] R µ Typeset by FoilTEX 7