Let X be any partially ordered set, R any commutative ring, and T = I ∗ ( X , R ) the weak incidence algebra of X over R . Let Z be a finite nonempty subset of X , L ( Z ) = { x ∈ X : x ≤ z    for some    z ∈ Z } , and M = T e z . Various chain conditions on M are investigated. The results so proved are used to construct some classes of right perfect rings that are not left perfect.