The reduced ring order and lower semi-lattices

TL;DR

This paper investigates the conditions under which reduced rings form lower semi-lattices with respect to a natural partial order, exploring examples from specific classes of rings and topological spaces.

Contribution

It characterizes when reduced rings are lower semi-lattices in the reduced ring order and studies liftings of orthogonal sets over surjective homomorphisms.

Findings

Certain classes of rings, like weakly Baer rings, form lower semi-lattices.

Topological spaces like locally connected and basically disconnected spaces yield rings that are lower semi-lattices.

Countable orthogonal sets can be lifted over surjective ring homomorphisms.

Abstract

Every reduced ring $R$ has a natural partial order defined by $a\le b$ if $a^2=ab$; it generalizes the natural order on a boolean ring. The article examines when $R$ is a lower semi-lattice in this order with examples drawn from weakly Baer rings (pp-rings) and rings of continuous functions. Locally connected spaces and basically disconnected spaces give rings $C(X)$ which are such lower semi-lattices. Liftings of countable orthogonal (in this order) sets over surjective ring homomorphisms are studied.