Let R be a left Artinian ring, and M a faithful left R-module such that no proper submodule or homomorphic image of M is faithful. If R is local, and socle(R) is central in R, we show that length(M=J(R)M) + length(socle(M)) ≤ length(socle(R)) + 1. If R is a finite-dimensional algebra over an algebraically closed field, but not necessarily local or having central socle, we get an inequality similar to the above, with the length of socle(R) interpreted as its length as a bimodule, and the summand +1 replaced by the Euler characteristic of a graph determined by the bimodule structure of socle(R). The statement proved is slightly more general than this summary; we examine the question of whether much stronger generalizations are possible. If a faithful module M over an Artinian ring is only assumed to have one of the above minimality properties - no faithful proper submodules, or no faithful proper homomorphic images - we find that the length of M=J(R)M in the former case, and of socle(M) in the latter, is ≤ length(socle(R)). The proofs involve general lemmas on decompositions of modules.

If L is a finite lattice, we show that there is a natural topological lattice structure on the geometric realization of its order complex Δ(L) (definition recalled below). Lattice-theoretically, the resulting object is a subdirect product of copies of L. We note properties of this construction and of some variants, and pose several questions. For M3 the 5–element nondistributive modular lattice, Δ(M3) is modular, but its underlying topological space does not admit a structure of distributive lattice, answering a question of Walter Taylor. We also describe a construction of “stitching together” a family of lattices along a common chain, and note how Δ(M3) can be regarded as an example of this construction.

The Boolean ring B of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (for example, 0) that are closed under the natural metric, but has no prime ideal closed under that metric; hence closed radical ideals are not, in general, intersections of closed prime ideals. Moreover, B is known to be complete in its metric. Together, these facts answer a question posed by J. Gleason. From this example, rings of arbitrary characteristic with the same properties are obtained. The result that B is complete in its metric is generalized to show that if L is a lattice given with a metric satisfying identically either the inequality d(x ∨ y, x ∨ z) ≤ d(y, z) or the inequality d(x ∧ y, x ∧ z) ≤ d(y, z), and if in L every increasing Cauchy sequence converges and every decreasing Cauchy sequence converges, then every Cauchy sequence in L converges; that is, L is complete as a metric space. We show by example that if the above inequalities are replaced by the weaker conditions d(x, x∨y) ≤ d(x, y), respectively d(x, x∧y) ≤ d(x, y), the completeness conclusion can fail. We end with two open questions.