Summary: Rings and Algebras Problem set #8: Solutions Nov. 10, 2011.
1. A morphism in a category C is a monomorphism if fx = fy implies x = y for morphisms x, y
in C. Similarly, g is an epimorphism if xg = yg implies x = y.
a) Describe monomorphisms and epimorphisms in the category C S (the category with one
object and a prescribed monoid S as the only Homset).
b) Give an example of a monoid for which there are morphism in C S which are both monomor
phisms and epimorphisms but which are not isomorphisms. (Recall that a morphism f is
an isomorphism if there is a morphism g such that fg and gf both exist and are equal to
the corresponding identity map.
Solution.a) Monomorphisms are the left cancellable, while epimorphisms are the right cancellable elements
of S. b) For the additivi monoid of N all elements are both monomorphisms and epimorphisms since both
cancellation laws are valid in N + . On the other hand only 0 is an isomorphism.
2. Determine whether the notions of monomorphisms and injective maps (or epimorphisms and
surjective maps) coincide in the following categories:
a) AB; b) RING; c) TOP; d*) GRP.
Solution.It is an easy exercise to show that whenever the objects of a category are sets and morphisms in the
category are set maps (i. e. we have a concrete category) then injective maps are monomorphisms. Similarly,
under the same circumstances, surjective maps are epimorphisms. Next, if f : A # B is not an injective map in
AB, GRP or RING then the inclusion i : Ker f # A is nonzero, but f # i = f # 0. Hence noninjective maps are
not monomorphisms, i. e. in all these categories injective maps and monomorphisms are the same. Similarly,