MTE-6-AST-2004
... For any three subsets A, B, C of a set U, A C if and only if A Bc C. The set of all mappings from {1, 2, , n} to itself form a group with respect to composition of maps. For any two elements a, b of a group G, o(ab) = o(ba). The set of elements of GL2 (R) whose orders divide a fixed numbe ...
... For any three subsets A, B, C of a set U, A C if and only if A Bc C. The set of all mappings from {1, 2, , n} to itself form a group with respect to composition of maps. For any two elements a, b of a group G, o(ab) = o(ba). The set of elements of GL2 (R) whose orders divide a fixed numbe ...
INTRODUCTION TO COMMUTATIVE ALGEBRA MAT6608
... For example, let R = k[X1 , . . . , Xn ] be the polynomial ring and Ri the collection of homogeneous polynomials of degree i, for the usual notion of degree and R+ is the maximal ideal of polynomials with zero constant term. A module (or ideal) of R is called a graded module, if there is a k-vector ...
... For example, let R = k[X1 , . . . , Xn ] be the polynomial ring and Ri the collection of homogeneous polynomials of degree i, for the usual notion of degree and R+ is the maximal ideal of polynomials with zero constant term. A module (or ideal) of R is called a graded module, if there is a k-vector ...
H9
... (a) φ : R[x] −→ R given by f (x) 7→ f (0). (b) φ : R[x] −→ R given by f (x) 7→ f (3). (c) φ : Z[x] −→ Z/5Z given by f (x) 7→ f (0) (mod 5). (d) φ : Z[x] −→ Z/5Z given by f (x) 7→ f (3) (mod 5). Here “describe” means give a criterion in terms of the coefficients of f (x) = a0 + a1 x + a2 x2 + · · · + ...
... (a) φ : R[x] −→ R given by f (x) 7→ f (0). (b) φ : R[x] −→ R given by f (x) 7→ f (3). (c) φ : Z[x] −→ Z/5Z given by f (x) 7→ f (0) (mod 5). (d) φ : Z[x] −→ Z/5Z given by f (x) 7→ f (3) (mod 5). Here “describe” means give a criterion in terms of the coefficients of f (x) = a0 + a1 x + a2 x2 + · · · + ...
HOMEWORK # 9 DUE WEDNESDAY MARCH 30TH In this
... integer n. Thus x + I 6= 0 + I but x + I = 0 + I. However, (x + I) = x + I = 0 + I = 0R/I proving that x + I is a non-zero nilpotent element. 2. Suppose that x ∈ R is a nilpotent element. Prove that 1 + x is a unit. Solution: Suppose xn = 0 for some integer n > 0. Then (1 + x)(1 − x + x2 − x3 + x4 − ...
... integer n. Thus x + I 6= 0 + I but x + I = 0 + I. However, (x + I) = x + I = 0 + I = 0R/I proving that x + I is a non-zero nilpotent element. 2. Suppose that x ∈ R is a nilpotent element. Prove that 1 + x is a unit. Solution: Suppose xn = 0 for some integer n > 0. Then (1 + x)(1 − x + x2 − x3 + x4 − ...
Section V.27. Prime and Maximal Ideals
... Examples 27.1 and 27.4. Consider the ring Z, which is an integral domain (it has unity and no divisors of 0). Then pZ is an ideal of Z (see Example 26.10) and Z/pZ is isomorphic to Zp (see the bottom of page 137). We know that for prime p, Zp is a field (Corollary 19.12). So a factor ring of an inte ...
... Examples 27.1 and 27.4. Consider the ring Z, which is an integral domain (it has unity and no divisors of 0). Then pZ is an ideal of Z (see Example 26.10) and Z/pZ is isomorphic to Zp (see the bottom of page 137). We know that for prime p, Zp is a field (Corollary 19.12). So a factor ring of an inte ...
Regular local rings
... /mi+1 is a is a finite dimensional k-vector space, and `A (i) = P each i,j m j+1 dim(m /m ) : j ≤ i. It turns out that there is a polyonomial pA with rational coefficients such that pA (i) = `A (i) for i sufficiently large. Let dA be the degree of pA . The main theorem of dimension theory is the fol ...
... /mi+1 is a is a finite dimensional k-vector space, and `A (i) = P each i,j m j+1 dim(m /m ) : j ≤ i. It turns out that there is a polyonomial pA with rational coefficients such that pA (i) = `A (i) for i sufficiently large. Let dA be the degree of pA . The main theorem of dimension theory is the fol ...
Algebra I
... placing two previously generated polynomial expressions into the blanks of the addition operator ( + ) or the multiplication operator ( × ). Monomial: A monomial is a polynomial expression generated using only the multiplication operator ( × ). Monomials are products whose factors are numerical expr ...
... placing two previously generated polynomial expressions into the blanks of the addition operator ( + ) or the multiplication operator ( × ). Monomial: A monomial is a polynomial expression generated using only the multiplication operator ( × ). Monomials are products whose factors are numerical expr ...
Completeness and Model
... The similarity of this result to Result 1.5 is no coincidence. Note, however, that Result 1.4 applies to the concept of an algebraically closed field of specified characteristic, not to the concept of an algebraically closed field in general. If we add to the concept of an algebraically closed field ...
... The similarity of this result to Result 1.5 is no coincidence. Note, however, that Result 1.4 applies to the concept of an algebraically closed field of specified characteristic, not to the concept of an algebraically closed field in general. If we add to the concept of an algebraically closed field ...