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 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 CS (the category with one object and a prescribed monoid S as the only Hom-set). b) Give an example of a monoid for which there are morphism in CS 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 non-zero, but f i = f 0. Hence non-injective maps are Collections: Mathematics