If R is a commutative ring with identity and ≤ is defined by letting a ≤ b mean ab = a or a = b, then (R,≤) is a partially ordered ring. Necessary and sufficient conditions on R are given for (R,≤) to be a lattice, and conditions are given for it to be modular or distributive. The results are applied to the rings Zn of integers mod n for n ≥ 2. In particular, if R is reduced, then (R,≤) is a lattice iff R is a weak Baer ring, and (R,≤) is a distributive lattice iff R is a Boolean ring, Z3, Z4, Z2[x]/x2Z2[x], or a four element field.
© 1999 Charles University in Prague
Henriksen, Melvin, and F. A. Smith. "The Bordalo order on a commutative ring." Commentationes Mathematicae Universitatis Carolinae 40.3 (1999): 429-440.