Sketched Solutions to Exercises of Chapter
نویسنده
چکیده
4.3. (i) Since HomA(A (I),M) ∼= M (I) for a set I and ⊕ is exact, it is clear that HomA(A (I),−) is exact, i.e. A(I) is projective. (ii) “Only if” part: Let A(P ) = ⊕ p∈P Ap be the free A-module indexed by P and ψ : A (P ) → p be the A-homomorphism such that ψ(1p) = p. Then ψ is surjective. If P is projective, then ψ has a section, i.e. an A-homomorphism φ : P → A(P ) such that ψ ◦ φ = idP . This shows P is a direct summand of A(P ). “If” part: If P ⊕K is free, then by (i) P ⊕K is projective, hence HomA(P ⊕K,−) = HomA(P,−)⊕ HomA(K,−) is exact. This implies HomA(P,−) is exact, hence P is projective.
منابع مشابه
Equation Chapter 1 Section 1 Analytical Solutions for Radially Functionally Graded Annular Plates
A closed-form solution for deflections and stresses in an annular thin plate of radially functionally graded material under transverse uniform pressure loading is presented. The small displacement theory of elasticity is assumed in the present work. Young’s modulus of the material is taken in the form of a simple power law to vary in the radial direction with an arbitrary exponent showing heter...
متن کاملComputer system security - Basic Concepts and Solved Exercises
This book is a collection of eight chapters and 106 solved exercises. Each chapter proposes an introduction to a generic problem encountered in computer security systems. After the introduction, the authors propose a set of exercises. Of course, the authors also reveal the succinct corresponding solutions. In a simplified summary, each chapter proposes a lesson, the examination and the correcte...
متن کامل