1. Chapter 4, §3: 5. I and A are additive subgroups, and the intersection of subgroups is a subgroup, so that I ∩A is an additive subgroup of R, whence of A. Now suppose that a ∈ A and i ∈ I ∩ A. Then a ∈ R and I is an ideal of R, so that ai ∈ I. On the other hand, A is a subring of R, so that ai ∈ A as i and a are in A. Thus ai ∈ I ∩ A. It follows that I ∩ A is an ideal. 6. As in the previous ...
