identity. All modules are left modules unless otherwise specified.

1. Let R be a commutative ring and M R Mod. Prove that M has the

structure of a bimodule in R ModR, with right module action given by

x r = rx for any r R, x M. What goes wrong with the argument when

R is noncommutative?

2. Let A, B be objects of an arbitrary category C, and let f A B and

g B A be morphisms in C, such that g f = 1A. Show that f is monic

and g is epic.

3. Let C be an abelian category. Let f A B be a morphism. Let k be a

morphism that is a kernel of f, and let c be a morphism that is a cokernel

of f. Prove that k is monic and c is epic.

4. Let C be an abelian category. Let f A B be a morphism. Let c =

cok(ker f) A C (the coimage of f). Prove that there is a unique map

i C B such that f = i c, and prove that i = ker(cok f).