15. Consider the mapping from M,(Z) into Z given by a b d -> a . Prove or disprove that this is a ring homomorphism. 16. Let R = 1. ab, cEZ . Prove or disprove that the map- ping a b C . - a is a ring homomorphism. 17. Is the mapping from Z to Z30 given by x - 6x a ring homomor- phism? Note that the image of the unity is the unity of the image but not the unity of Z30- 18. Is the mapping from Z1, to Z,, given by x -> 2x a ring homomor- phism? 19. Describe the kernel of the homomorphism given in Example 3. 20. Recall that a ring element a is called an idempotent if a2 = a. Prove that a ring homomorphism carries an idempotent to an idempotent. 21. Determine all ring homomorphisms from Zo to Zo. Determine all ring homomorphisms from Z2, to Z30- 22. Determine all ring isomorphisms from Z to itself. 23. Determine all ring homomorphisms from Z to Z. 24. Suppose o is a ring homomorphism from ZOO Z into Z O Z. What are the possibilities for d((1, 0))? 25. Determine all ring homomorphisms from ZO Z into Z O Z. 26. In Z, let A = (2) and B = (8). Show that the group A/B is isomor- phic to the group Z, but that the ring A/B is not ring-isomorphic to the ring Z4. 27. Let R be a ring with unity and let o be a ring homomorphism from R onto S where S has more than one element. Prove that S has a unity. 28. Show that (Z Z)/((a) D (b)) is ring-isomorphic to Z. D Z. 29. Determine all ring homomorphisms from Z O Z to Z. 30. Prove that the sum of the squares of three consecutive integers can- not be a square. 31. Let m be a positive integer and let n be an integer obtained from m by rearranging the digits of m in some way. (For example, 72345 is15. Consider the mapping from M,(Z) into Z given by a b d -> a . Prove or disprove that this is a ring homomorphism. 16. Let R = 1. ab, cEZ . Prove or disprove that the map- ping a b C . - a is a ring homomorphism. 17. Is the mapping from Z to Z30 given by x - 6x a ring homomor- phism? Note that the image of the unity is the unity of the image but not the unity of Z30- 18. Is the mapping from Z1, to Z,, given by x -> 2x a ring homomor- phism? 19. Describe the kernel of the homomorphism given in Example 3. 20. Recall that a ring element a is called an idempotent if a2 = a. Prove that a ring homomorphism carries an idempotent to an idempotent. 21. Determine all ring homomorphisms from Zo to Zo. Determine all ring homomorphisms from Z2, to Z30- 22. Determine all ring isomorphisms from Z to itself. 23. Determine all ring homomorphisms from Z to Z. 24. Suppose o is a ring homomorphism from ZOO Z into Z O Z. What are the possibilities for d((1, 0))? 25. Determine all ring homomorphisms from ZO Z into Z O Z. 26. In Z, let A = (2) and B = (8). Show that the group A/B is isomor- phic to the group Z, but that the ring A/B is not ring-isomorphic to the ring Z4. 27. Let R be a ring with unity and let o be a ring homomorphism from R onto S where S has more than one element. Prove that S has a unity. 28. Show that (Z Z)/((a) D (b)) is ring-isomorphic to Z. D Z. 29. Determine all ring homomorphisms from Z O Z to Z. 30. Prove that the sum of the squares of three consecutive integers can- not be a square. 31. Let m be a positive integer and let n be an integer obtained from m by rearranging the digits of m in some way. (For example, 72345 is