Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 2512. Let R be the ring of 2 × 2 matrices of the form( ) a b,0 0where a, b ∈ R. Show that although R is a ring that has no identity, we canfind a subring S of R with an identity.3. List or characterize all of the units in each of the following rings.(a) Z 10(b) Z 12(c) Z 7(d) M 2 (Z), the 2 × 2 matrices with entries in Z(e) M 2 (Z 2 ), the 2 × 2 matrices with entries in Z 24. Find all of the ideals in each of the following rings. Which of these ideals aremaximal and which are prime?(a) Z 18(b) Z 25(c) M 2 (R), the 2 × 2 matrices with entries in R(d) M 2 (Z), the 2 × 2 matrices with entries in Z(e) Q5. For each of the following rings R with ideal I, give an addition table and amultiplication table for R/I.(a) R = Z and I = 6Z(b) R = Z 12 and I = {0, 3, 6, 9}6. Find all homomorphisms φ : Z/6Z → Z/15Z.7. Prove that R is not isomorphic to C.8. Prove or disprove: The ring Q( √ 2 ) = {a + b √ 2 : a, b ∈ Q} is isomorphic tothe ring Q( √ 3 ) = {a + b √ 3 : a, b ∈ Q}.9. What is the characteristic of the field formed by the set of matrices{( ) ( ) ( ) ( )}1 0 1 1 0 1 0 0F =, , ,0 1 1 0 1 1 0 0with entries in Z 2 ?10. Define a map φ : C → M 2 (R) by( a bφ(a + bi) =−b a).Show that φ is an isomorphism of C with its image in M 2 (R).