MATH20212 Algebraic Structures 2
Take Away Test
1. Let S = {a + b√-2 | a, b ∈ Q}. Show that S is a subring of C. Is S a field? Explain your answer. [5 marks]
2. Write down a zero divisor in M3 (Z).
Consider the matrix equation AX = A, where A is a zero divisor in M3 (Z). Explain why there is more than one solution for X ∈ M3 (Z).
Find two solutions for X ∈ M3 (Z) using your zero divisor as the matrix A. [5 marks]
3. Consider the map θ : Z10 → Z5 × Z2 , defined by θ([a]10 ) = ([a]5, [a]2 ) for all [a]10 ∈ Z10 . Explain why θ is a well-defined function.
Show that θ is a ring homomorphism. Is θ injective? Explain your answer. [6 marks]
4. Let I = {p(X) ∈ R[X] | p(3) = 0}. Show that I is an ideal of R[X]. By finding a suitable generator, write I as a principal ideal.[4 marks]