Q.5) Prove (1) The trivial submodule 0 is the only small submodule in Zz (2) If MN is a homomorphism and if A is small in M then (A) is small in (3) Let ABS M. If B is small in M then M is small in A

Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].

Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .

Let be as described in the proof of Theorem. Give a specific example of a positive element of .

For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={ xifx0xif0x Prove that | x |=| x | for all xD. Prove that | x |x| x | for all xD. Prove that | xy |=| x || y | for all x,yD. Prove that | x+y || x |+| y | for all x,yD. Prove that | | x || y | || xy | for all x,yD.

Describe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].

Show that (p(x), q(x)) = p(0)q(0) +p(1)q(1) +p(2)q(2) defines an inner product on P2.

1. (a) For the invariant theory connected to the general linear model, find g. (b) Show that ğ : 0₁ → 9₁ and 9: 0₂ ā (c) Show that 8² = 0 ₂ for all g. Σ7_₁ ? is the maximal invariant in © = {§1, ..., Ék, 0²}.

2. Prove that f: (R × R, +) → (R,+) given by f(x, y) = x + y is a homomorphism. Is g: (R × R, +) → (R, +) given by g(x, y) = x.y a homomorphism? Explain your answer. (Hint: Check whether f((a, b) + (c,d)) = f(a,b) + f(c,d).)

2. Does there exist a linear transformation T: R4 → R with nullspace X2 ker (T) = -(13-- = X3 X4 € R¹ : £1 = 3x2, x3 = x4 If yes, give an example of such a transformation. If not, give a proof that no such transfor- mation can exist.