Let I be a non-empty index set with a partial order<=. Assume that I is a directed set, that is, that for any pair i,j in I there is a,k in J such that i<=k and j<=k. Suppose that for every pair of indices i,j with i<=j ther is a map p_ij: A_i->A_j such that p_jkp_ij=p_ik whenever i<=j<=k and p_ii=1 for all i in I. Let B be the disjoint union of all A_i. Define a relation ~ on B by setting a~b if there exists k such that i<=k and j<=k and p_ik(a)=p_jk(b) where a is in A_i and b is in A_j.
a) Show that ~ is an euivalence relation on b. (The set of equivalence classes is called the direct or inductive limit of the directed system {A_i} and is denoted lim->A_i. For the rest of the problem let A=lim->A_i
b) Let x' denote the class of x in A and define p_i:A_i->A by p_i(a)=a'. Show that if each p_ij is injective so is p_i for all i so that we may then identify each A_i as a subset of A.
c)Assume all p_ij are group homomorphisms. For a in A_i, b in A_j show that the operation a'+b'=(P-ik(a)+p_jk(b))' where k is any index with i,j<=k, is well defined and makes A into an abelian group. Deduce that the maps p_i in part (b) of the problem are group homomorphisms from A_i to A.
d)Show that if all A_i are commutative rings with 1 and all p_ij are rings homo. that send 1 to 1, then A may likewise be given the structure of a commutative ring with 1 such that all p_i are ring homomorphisms.
e)Under the hypothesis of part (c) prove that the direct limit has the following universal mapping property: if C is any abelian group such that for each i there is a homomorphism phi_i:A_i ->C with phi_i=phi_jphi_ij whenever i<=j, then there is a unique group homomorphism phi: A->C such that phip_i=phi_i