# Unique Ring Homomorphisms

Let F be a field. Let . Prove that there exists a unique ring homomorphism such that .

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Ring Theory : Direct Products, Mappings and Homomorphisms

Let I be a non-empty index set with a partial order <=, and A_i be a group for all i in I. Suppose that for every pair of indices i,j there is a map phi_ij:A_j ->A_i such that phi_jiphi_kj= phi_ki wheneveri<=j<=k, and phi_ii=1 for all i in I. Let P be the subset of elements(a_i) with i from I in the direct product D of A_i such that phi_ij(a_j)=a_i whenever i<=j. P is called the inverser or projective limit of the system {A_i} and is denoted lim<-A_i.

a) Assume all phi_ji are group homomorphisms. Show that P is a subgroup of the direct product of the groups A_i.

b) Assume I=Z+ and phi_ji are group homomorphisms. For each i in I let phi_i:P->A_i be the projection of P onto its i-th component.Show that if each phi_ji is surjective, then so is phi_i, for all i so that each A_i is a quotient group of P

c)Show that if all A_i are commutative rings with 1 and all phi_ji are ring homomorphisms that send 1 to 1, then A may likewise be given the structure of a commutative ring with 1 such that all phi_i are ring homomorphisms.

d)Assume all phi_ji are group homomorphisms. Prove that the inverse limit has the following universal mapping property: if K is any group such that for each i in I there is a homomorphism pi_i:K->A_i with pi_i=phi_jipi_j whenever i<=j, then there is a unique homomorphism pi: K->P such that phi_ipi=pi_i for all i in I