Torsion-Theoretic Krull Dimension

Let ϕ = U_-1 ⊆ U₀ ⊆ … be the ascending chain of subsets of R-sp defined as follows:

U₀ is the set of all minimal elements of R-sp;

If i is not a limit ordinal then U_i = {Π ∈ R-sp | π > Π′ ∈ R-sp ⇒ π′ ∈ U_i-1};

If i is a limit ordinal then U_i = ∪_j<i U_j.