Math 515 homework #6 Due date: 19980206 D. J. Bernstein 19980130

Operations on Z-submodules of a ring. Let A be a Z-algebra, i.e., a ring. Let M and N be Zsubmodules of A, i.e., subgroups of A under 0, −, +. The sum M + N = {m + n : m ∈ M, n ∈ N} is the smallest Z-submodule of A containing M and N. The product MN is the smallest Z-submodule of A containing {mn : m ∈ M, n ∈ N}, namely the set of finite sums m_{1}n_{1} + ⋯ + m_{k}n_{k}. In the following exercise, M, N, and T are Z-submodules of a ring A. Exercise 14. Prove that T (M + N) = T M + T N. Other useful identities: M + N = N + M; MN = NM; M + (N + T) = (M + N) + T; M(NT) = (MN)T.

Operations on R-submodules of an R-algebra. In the following exercise, R is a ring, and A is an Ralgebra. Exercise 15. Let M be a Z-submodule of A. Let N be an R-submodule of A. Prove that MN is an R-submodule of A. Thus R-submodules of an R-algebra are closed under Z-submodule operations; there is no need for a separate definition of R-submodule operations. In particular, if M and N are ideals of a ring R, then MN and M +N are ideals of R.