## 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

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