11. Find all possible ring homomorphisms from A to B: (a) A = Z10, (b) A...
Consider ring homomorphisms from Z to Z/nZ List all ring homomorphisms from Z to Z/15Z. In each case, describe the kernel and the mage
Every ring in this test is commutative with 1 and 1 0 1. Which of the followings are prime ideals of Z? (Separate your answers by commas.) A. ( B. (2). C. (9). D. (111). E. (101) 2. Which of the followings are ring homomorphisms? (Separate your answers by commas.) A.φ: Z → Z, defined by (n) =-n for all n E Z B. ф: Z[x] Z, defined by ф(p(z)) p(0) for all p(z) E Z[2] C. : C C....
a. If A and B are R-modules, then the set HomR(A,B) of all R-module homomorphisms A B is an abelian group with f + g given on a ∈ A by (f + g)(a) = f(a) + g(a) ∈ B. The identity element is the zero map. b. HomR(A,A) is a ring with identity, where multiplication is composition of functions. HomR(A,A) is called the endomorphism ring of A. c. A is a left HomR(A,A)-module with fa defined to be f(a)(a...
two questions,please! 11. Let φ: R → S be a ring hornomorphism. Prove that if R is a field and φ(R)メ(0), then φ(R) is a field 12, Give an example where φ: R → s is a surjective homomorphisms of unitary rings and the restricted 11. Let φ: R → S be a ring hornomorphism. Prove that if R is a field and φ(R)メ(0), then φ(R) is a field 12, Give an example where φ: R → s is a...
Definition A: Let R be ring and r e R. Then r is called a zero-divisor in Rifr+0r and there exists SER with s # OR and rs = OR. Exercise 1. Let R be a ring with identity and f € R[2]. Prove or give a counter example: (a) If f is a zero-divisor in R[x], then lead(f) is a zero-divisor in R. (b) If lead(f) is a zero- divisor in R[x], then f is a zero-divisor in R[2]....
For each part below, list all homomorphisms with the given domain and codomain a) Domain Zis and codomain Zs (b) Domain Zu and codomain Zo c) Domain and codomain both Z4 (d) Domain C4 and codomain V (e) Domain and codomain both Va VA. For each part below, list all homomorphisms with the given domain and codomain a) Domain Zis and codomain Zs (b) Domain Zu and codomain Zo c) Domain and codomain both Z4 (d) Domain C4 and codomain...
(a) Let C be an elliptic curve. Define the endomorphism ring of C to be 6.16. End(C) endomorphisms CC) Note that this is a little different from the endomorphism ring of C considered as an abelian group, because we are not taking all group homomorphisms from C to itself, but only those defined by rational functions. In other words, End(C) is the set of algebraic endomorphisms of C. Prove that the addition and multi plication rules make End(C) into a...
Algebraic structures 1. Consider the ring M = {Ia al: a, b, c, d e Z2} under entry-wise addition and standard matrix multiplication. a. What are the units of this ring? b. Determine whether or not it is an integral domain. 2. Consider the ring Z * ZZ under component-wise addition and multiplication. a. What are the units of this ring? b. Let I = ( (2,1,1)) and J = ( (1,3,1)) be principal ideals. Show that their intersection is...
All the charge in a ring of charge Q is the same distance r from a point P on the ring axis. a) Electric charge Q is distributed uniformly around a thin ring of radius a (Fig. 23.20). Find the potential at a point P on the ring axis at a distance x from the center of the ring. b) Find the electric field at P using the appropriate denotative relationships
(b) (6 marks) Find a decomposition of the module Z/10Z over the ring Z into a direct sum of indecomposable modules. (b) (6 marks) Find a decomposition of the module Z/10Z over the ring Z into a direct sum of indecomposable modules.