MATH 611 Due Wed Oct 7, 1992

1. Let N be a submodule of M and let θ: M → M/N be the quotient map (canonical epimorphism). Prove that there is a one-to-one correspondence between the submodules of M/N and the submodule of M containing N, where a submodule L of M/N corresponds to the submodule θ^{−1}(L) of M. note: The notation does not refer to a map θ^{−1} . Unless N = 0 there is no inverse to θ (why?). Instead, by definition

θ^{−1}(L) = { m ∈ M | θ(m) ∈ L }.

2. Let R be a ring and let R_{1} and R_{2} be additive subgroups of R closed under multiplication such that R = R_{1} + R_{2} . (note: According to our conventions, we do not refer to R_{1} and R_{2} as subrings of R since they probably (in fact, almost certainly!) will not contain 1.) Prove that the following conditions are equivalent:

(1) R_{1} ∩ R_{2} = 0 and for all r_{1}, r′_{1} ∈ R_{1} and r_{2}, r′_{2} ∈ R_{2} ,

(r_{1} + r_{2})(r′_{1} + r′_{2}) = r_{1}r′_{1} + r_{2}r′_{2}.

(2) R_{1}R_{2} = R_{2}R_{1} = 0. I.e. for all r_{1} ∈ R_{1} and r_{2} ∈ R_{2} , r_{1}r_{2} = r_{2}r_{1} = 0.

(3) R_{1} and R_{2} are (two-sided) ideals of R and R_{1} ∩ R_{2} = 0.

(4) There exist e_{1}, e_{2} ∈ R such that R_{1} = Re_{1} , R_{2} = Re_{2} and

e_{1}^{2} = e_{1}, e_{2}^{2} = e_{2}, e_{1} + e_{2} = 1, e_{1}e_{2} = e_{2}e_{1} = 0

and for all r ∈ R, re_{1} = e_{1}r, re_{2} = e_{2}r.

(This last condition means that e_{1} and e_{2} are in the center of R.)

3. Let F be a skewfield and V a vector space over F with dim_{F} V = n < ∞. Let α ∈ End_{F} V . We say that α is nilpotent if α^{r} = 0 for some positive integer r.

a) Prove that α is nilpotent if and only if α^{n} = 0, where n = dim_{F} V .

b) Prove that α is nilpotent if and only if there exists a basis for V such that the matrix for α with respect to the basis is strictly upper triangular (i.e. has zeros on and below the main diagonal).

4. Let M be an A-module and α ∈ End_{A} M. Suppose there exists a positive integer n such that α^{n}(M) = α^{n+1}(M) and Ker α^{n} = Ker α^{n+1} . Prove that M = α^{n}(M) ⊕ Ker α^{n}.