Melvin Henriksen, F.A. Smith
The Bordalo order on a commutative ring

Comment.Math.Univ.Carolinae 40,3 (1999) 429-440.

Abstract:If $R$ is a commutative ring with identity and $\leq $ is defined by letting $a\leq b$ mean $ab=a$ or $a=b$, then $(R,\leq )$ is a partially ordered ring. Necessary and sufficient conditions on $R$ are given for $(R,\leq )$ to be a lattice, and conditions are given for it to be modular or distributive. The results are applied to the rings $Z_{n}$ of integers mod $n$ for $n\geq 2$. In particular, if $R$ is reduced, then $(R,\leq )$ is a lattice iff $R$ is a weak Baer ring, and $(R,\leq )$ is a distributive lattice iff $R$ is a Boolean ring, $Z_{3},Z_{4}$, $Z_{2}[x]/x^{2}Z_{2}[x]$, or a four element field.

Keywords: commutative ring, reduced ring, integral domain, field, connected ring, Boolean ring, weak Baer Ring, regular element, annihilator, nilpotents, idempotents, cover, partial order, incomparable elements, lattice, modular lattice, distributive lattice
AMS Subject Classification: 03G10, 06A06, 11A07, 13A99