Question

) In the Extra Problem on PS 2 we defined a map φ:M2,2(Z) → M2,2(Z2) by the formula φ a b c d = a mod (2) b mod (2) c mod (2) d mod (2)

On PS 2 you showed that φ is a ring homomorphism and that ker(φ) = 2a 2b 2c 2d a, b, c, d ∈ Z We know the kernel of any ring homomorphism is an ideal. Let I = ker(φ).

(a) (6 points) The target ring M2,2(Z2) is a finite ring. Find (or describe) all of the elements of this ring. What is | M2,2(Z2)|?

(b) (6 points) Show φ is an onto ring homomorphism. Since you have already shown φ is a ring homomorphism, you only need to show φ M2,2(Z) = M2,2(Z2).

(c) (6 points) Use Parts (a) and (b) and the First Isomorphsm Theorem to explain why M2,2(Z)/I is a finite ring and determine |M2,2(Z)/I|.

(d) (6 points) Determine whether M2,2(Z2) is a commutative ring. What does this say about the quotient ring M2,2(Z)/I? Why?

(e) (6 points) Does M2,2(Z2) have zero divisors? Prove your answer. What does this say about the quotient ring M2,2(Z)/I? Why?

Answer 1