Math 202 -assignment 1
نویسنده
چکیده
a. N1 = {(i1, i2, . . . , in) : ik ∈ Ik for all k ∈ {1, 2, . . . , n}} and b. N2 = {(x1, x2, . . . , xn) : ∑n k=1 xk = 0}. Proof. To prove (a), it suffices to show, by Proposition 1, that N1 is nonempty and x + ry ∈ N1 for all r ∈ R and all x, y ∈ N1. For the first condition, (0, 0, . . . , 0) ∈ N1 since Ik is a subgroup of R containing the additive identity 0 for all k ∈ {1, 2, . . . , n}. That is, N1 is nonempty. For the second condition, let x = (ik)k∈Z+ , y = (yk)k∈Z+ ∈ N1 and let r ∈ R. Then, by definition of addition and scalar multiplication,
منابع مشابه
Math 202 -assignment 7
a. N1 = {(i1, i2, . . . , in) : ik ∈ Ik for all k ∈ {1, 2, . . . , n}} and b. N2 = {(x1, x2, . . . , xn) : ∑n k=1 xk = 0}. Proof. To prove (a), it suffices to show, by Proposition 1, that N1 is nonempty and x + ry ∈ N1 for all r ∈ R and all x, y ∈ N1. For the first condition, (0, 0, . . . , 0) ∈ N1 since Ik is a subgroup of R containing the additive identity 0 for all k ∈ {1, 2, . . . , n}. Tha...
متن کاملMath 202 - Assignment 7 Solutions
Exercise 10.3.2. Let R be a commutative ring with identity. For all positive integers n and m, R ∼= R if and only if n = m. Proof. Let φ : R → R be an isomorphism of R-modules and let I E R be a maximal ideal. Then the map φ̄ : R → R/IR given by φ̄(α) = φ(α) is a morphism of R-modules. Moreover ker φ̄ = {α ∈ R | φ̄(α) = 0} = {α ∈ R | φ(α) ∈ IR} = φ−1(IRm) = IR. Therefore by the first isomorphism th...
متن کاملMath 202 - Assignment 6
Proof. The discriminant of x + 1 is D = 256 = 2. We have x + 1 ≡ (x + 1) (mod 2). Let p be an odd prime (so p D), and suppose the irreducible factors of x + 1 have degrees n1, n2, . . . , nk. By Corollary 41, the Galois group of x + 1 contains an element with cycle structure (n1, n2, . . . , nk). Since the Galois group of x +1 over Q is the Klein 4-group, in which every element has order dividi...
متن کاملThe Extent of Using Real Problems and Emphasizing Modelling in the 10th Grade Math Textbook
Modelling is emphasized heavily in the national math curriculum, yet the extent to which this concept is covered in the math textbooks is, while of great importance, unknown. To discover the extent to which this issue, and real problems in general, are covered in the 10th grade math textbooks, the content of the textbook for the ‘theoretical’ (“nazari”) branch was analyzed. In doing so, the vie...
متن کاملThe univalence conditions for a general integral operator
For analytic functions in the open unit disk, J. Becker (Math. Ann. 202(1973)) has given some univalent conditions. In the present paper, some extensions of Becker’s type are considered.
متن کامل