Consider the ring Z3 = {[0], [1], [2]} and the ring Z6 = {0, ., 4,...

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Consider the ring Z3 = {[0], [1], [2]} and the ring Z6 = {0, …., 4, 5}. Define g: Z3 → Z6 by g([0]) = Ō, g([1]) = 4 and g([2]) = 2. (1) Determine whether g is a ring homomorphism. (2) Determine Ker(g) explicitly, if g is a ring homomorphism. (3) Determine Im(g) explicitly, if g is a ring homomorphism. Hint. As Z3 is a finite ring, one could check the equations g(x + y) g(x) + g(y) and g(xy) = g(x)g(y) by exhausting all combinations of x, y ≤ Z3 {[0], [1], [2]}. To distinguish elements in Z3 from elements in Z6, we use [0], [1], [2]} to denote the elements of Z3. = = Consider the ring Z3 = {[0], [1], [2]} and the ring Z6 = {0, …., 4, 5}. Define g: Z3 → Z6 by g([0]) = Ō, g([1]) = 4 and g([2]) = 2. (1) Determine whether g is a ring homomorphism. (2) Determine Ker(g) explicitly, if g is a ring homomorphism. (3) Determine Im(g) explicitly, if g is a ring homomorphism. Hint. As Z3 is a finite ring, one could check the equations g(x + y) g(x) + g(y) and g(xy) = g(x)g(y) by exhausting all combinations of x, y ≤ Z3 {[0], [1], [2]}. To distinguish elements in Z3 from elements in Z6, we use [0], [1], [2]} to denote the elements of Z3. = =