The inverse limit of a family of R-modules is the dual notion of a direct limit and is characterized by the following mapping property. For a directed set I and a family of R-modules {M_i}_(i element I), let (M_i, σ_(j i)) be an inverse system. lim_⟵ M_i is some R-module with some homomorphisms σ_i, where for each i element I, i<=j σ_i :lim_⟵ M_i->M_i with the property σ_i = σ_(j i) °σ_j such that if there exists some R-module N with homomorphisms α_i, where for each i element I, i<=j α_i :N->M_i with the property α_i = σ_(j i) °α_j, then a unique homomorphism α:N->lim_⟵ M_i is induced and the above diagram commutes.
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