HW due Fri Feb 16

1a. Let A be an (R,S)-bimodule, and B an abelian subgroup that is both an R-submodule and a (right) S-submodule. Show that A/B is again naturally an (R,S)-bimodule.
(Don't worry about proving naturality.)

b. Let I,J be two-sided ideals of R. Show that the (R,R)-bimodule R/I @_R R/J is isomorphic to R/K for some other ideal K.

2. Let M > N be a pair of R-modules. If N and M/N are finitely generated, show M is also finitely generated.

3. Let I be a 2-sided ideal of R, and for any module M, let IM := { sum_j i_j m_j }.

a. Show that IM is an R-submodule.

b. Show that M/IM is naturally an R/I-module (not just an R-module).
(Don't worry about proving naturality.)

c. Figure out how to describe M/IM as a degenerate sort of "extension of scalars" (!).

d. If M a free module over R, show M/IM is a free module over R/I.

4. Let R be the (commuting) polynomial ring C[x,y] and I the ideal < x, y > generated by the variables. Show I is not a free module.