Ideals: Definitions & Examples

Transcription

1 Ideals: Definitions & Examples Defn: An ideal I of a commutative ring R is a subset of R such that for a, b I and r R we have a + b, a b, ra I Examples: All ideals of Z have form nz = (n) = {..., n, 0, n, 2n,...} All ideals of Z[i] have form αz[i] = (α) = {αβ β Z[i]} for α Z[i] Z[ 5] has ideals of form: αz[ 5] = (α) = {αβ β Z[ 5]} for α Z[ 5] (α 1, α 2 ) = {α 1 β 1 + α 2 β 2 β 1, β 2 Z[ 5]} for α 1, α 2 Z[ 5] Z[x] has the ideal (2, x) - all polynomials with even constant terms. Robert Campbell (UMBC) 5. Rings May 5, / 12

2 Ideals: 1847, FLT, etc Lamé s Proof : In 1847 Lamé presented a proof of Fermat s Last Theorem with a rough outline of: Given a p + b p = c p Factor as (a b)(a ξ p b)(a ξpb) 2... (a ξp p 1 b) = c p where ξ p = cos( 2π p ) + i sin( 2π p ) is a primitive pth root of unity and Z[ξ p] is the Cyclotomic integers. If {(a ξ k b)} are coprime then each of the (a ξ k b) must be a pth power. If they are not coprime, divide by the common factor and repeat (infinite descent). Gap: Proof assumes that if x, y are irreducible and xy = z n then x, y are nth powers - depends on unique factorization, but Z[ξ 23 ] does not have unique factorization. Solution: Ideal numbers. Robert Campbell (UMBC) 5. Rings May 5, / 12

3 Kummer & Dedekind Kummer (1847) proposed that unique factorization could be recovered by adding new elements to the ring - ideal numbers. Dedekind published (1876) the concept of ideal as a set of elements preserved under addition, negation and multiplication, which could be thought of as the set of multiples of an ideal number. Hilbert s Example: Consider the set S = {n 1(mod 4)} = {1, 5, 9,...} Closed under multiplication (ignore addition) Irreducibles are P = {5, 9, 13, 17, 21, 29,..., 49,...} 441 = (21)(21) = (9)(49) [over Z gcd(9, 21) = 3, gcd(21, 49) = 7, so 441 = (3 2 )(7 2 )] Kummer Ideal Number: Add 3 and 7 to the set (multiples?) Dedekind Ideal: Replace numbers with sets: [3] = {9, 21, 33, 45,...} [7] = {21, 49, 77,...} Robert Campbell (UMBC) 5. Rings May 5, / 12

4 Ideals & Ideal Numbers: An Example Example: Consider Z[ 5] 6 = (2)(3) = (1 + 5)(1 5) Ideals: Factor 3, 2, (1 + 5) and (1 5) as ideals (3) = (3, 5 + 1)(3, 5 1) (2) = (2, 5 + 1) 2 (1 + 5) = (2, 5 + 1)(3, 5 + 1) (1 5) = (2, 5 + 1)(3, 5 1) Ideals: adjoin 2 to the ring Robert Campbell (UMBC) 5. Rings May 5, / 12

5 Ideals: Generating Sets Defn: The ideal of R generated by set X is the intersection of all ideals of R containing X, I (X ) = {I X I, I ideal R}. (also denoted (X )) Prop: For X finite, I (X ) = { N i=0 β ix i β i R}. pf: I (X ) is closed under addition and multiplication by R, so it is an ideal. X I = I (X ) I as I must be closed. Robert Campbell (UMBC) 5. Rings May 5, / 12

6 Principal Ideals Defn: An ideal is principal if it has some generating set of one element, A = (α) Examples: All ideals of Z have form nz = (n) and are principal Which can also be written as (n) = (n, 3n, 5n) = (3n, 7n) All ideals of Z[i] have form αz[i] = (α) and are principal Z[ 5] has ideals of form: αz[ 5] = (α) = {αβ β Z[ 5]} for α Z[ 5] (principal) (α 1, α 2 ) = {α 1 β 1 + α 2 β 2 β 1, β 2 Z[ 5]} for α 1, α 2 Z[ 5] (non-principal) Z[x] has the ideal (2, x) - all polynomials with even constant terms. (non-principal) Robert Campbell (UMBC) 5. Rings May 5, / 12

7 PIDs Defn: A principal ideal domain (PID) is an integral domain, all of whose ideals are principal. Examples: Z is a PID F[x] is a PID Z[x] is not a PID (2, x) Z[x] is not principal Robert Campbell (UMBC) 5. Rings May 5, / 12

8 Algebra of Ideals Defn: Robert Campbell (UMBC) 5. Rings May 5, / 12

9 Prime & Maximal Ideals Defn: An ideal of R is prime if for any a, b R, if ab I, then either a I or b I Defn: An ideal I of R is maximal if I J for some ideal J implies J = I or J = R Prop: If I is a maximal ideal of R then I is a prime ideal of R. Prop: If I is a non-zero prime ideal of PID R then I is a maximal ideal of R. Examples: Z: (n) Z is maximal and prime iff n Z is prime Q[x]: (f (x)) [x] is maximal and prime iff f (x) Q[x] is irreducible Q[ 6]: (2, 6) Q[ 6] is prime Robert Campbell (UMBC) 5. Rings May 5, / 12

10 Polynomial Ideals Defn: An affine algebraic set S is the set of points in the vector space F n which satisfy a finite set of polynomial equations {f i (x 0,..., x n ) = 0}. Defn: An algebraic set S is reducible iff it can be expressed as the union of two proper subsets, each of which is an algebraic set, S = S 1 S 2 Defn: An algebraic variety is an irreducible algebraic subset. Examples: The union of curves {(x, y) y = x 2 + 1} {(x, y) y = 2x} is an algebraic set but not a variety Each curve, {(x, y) y = x 2 + 1} or {(x, y) y = 2x} is an algebraic variety Prop: The set of polynomials in F[x 0,..., x n ] which is satisfied by all points in some variety V is an ideal, I (V ). Prop: An algebraic variety V is irreducible iff I (V ) is a prime ideal. Robert Campbell (UMBC) 5. Rings May 5, / 12

11 Quotient Construction Lemma: The relation r 1 r 2 iff r 1 + I = r 2 + I is an equivalence relation. Defn: The quotient ring is R/I = {r + I }/, where addition and multiplication is inherited from R. Prop: R/I is a ring Examples: Z n = Z/nZ Robert Campbell (UMBC) 5. Rings May 5, / 12

12 Quotients: Prime & Maximal Ideals Prop: R/P is an integral domain iff P R is a prime ideal Prop: R/M is a field iff M R is a maximal ideal Robert Campbell (UMBC) 5. Rings May 5, / 12