In the following, R will denote an associate ring admitting a multiplicative
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).