1. (10+5 pts) (a) Let / and J be two left ideals of R and let It J = {xty : re l, ye J}. Prove that I + J is a left ideal of R. (b) Formulate and prove an analogous statement for two-sided ideals. 2. (15 pts) Let R be a commutative ring, a, b E R and let (a2b - a) be the principal ideal generated by a b - a. Prove that the image (ab) = ab + (a2b - a) of ab in R/(a2b - a) under the natural homomorphism 7 : R - R/(a2b - a) is an idempotent. 3. (a) (10 pts) Show that the ideal J = (2, X) of the ring Z[X] consists precisely of the polynomials from Z[X] with constant term even. (b) (10 pts) Show that the ideal J = (2, X) is a maximal ideal of the ring Z[X]. 4. (20 pts) Let R be a commutative ring. Show that the ideal (X) is a prime ideal of R[X] if and only if R is an integral domain. 5. (15 pts) Show that the element 3 is irreducible in R = Z[V-5]. 6. (15 pts) Show that the element 3 is not prime in R = Z[v-5]