Denote by Zm = {[0]m [1]m,..., [m - 1]m} the ring of integers modulo m. Consider the rings R = Z24 and S = Z4 × Z6. Let 0 RS be the map defined by ([x]24) = ([x]4, [4x]6). (b) Is a homomorphism of rings? Explain.

3. Determine whether the following mappings are ring homomorphisms: f: Q- Q defined by f(x)= |x| for all x e Q. a) b) c) g: CM₂ (R) defined by g (a + bi) = a b -b a h: Z[√2] → Z[√2] defined by h( a + b √2) =) a - b √2

Let D: Q[x] → Q[x] akx²+...+a₁x + að ↔ kakxk−1 +...+a₁ be the derivative map on rational polynomials. Is D a ring homomorphism? Is it an isomorphism?

Let R be a commutative ring with unity and c E R. The map R[X] → R[X] f(X) → R= f(X + c) is an isomorphism of rings.