1. Show that one side inverse is not enough for an element of a ring to have a multiplicative inverse. 2. Give an example of a Division ring which is not a field. 3. Let f: A → B be a ring homomorphism. Show by example that: a. Im(f) need not be an ideal of B. b. If M is a maximal ideal of B, then f-1 (M) need not be a maximal ideal of A. 4. Prove via a counter-example that if A is infinite, then an ideal of Пlies Ai doesn't necessarily have the form ПlieAli, where Ai are rings and I; is an ideal of A₁. 5. Let 1, 2, …, In be ideals of a ring A with I; + I; = A, Vi #j. Show that: " N²±14 = ±14. 6. Let f: A → B be a surjective ring homomorphism. If A is local and B is non zero, show that B is local.