The Bordalo Order on a Commutative Ring

Authors

Melvin Henriksen, Harvey Mudd CollegeFollow
Frank A. Smith, Kent State UniversityFollow

Document Type

Article

Department

Mathematics (HMC)

Publication Date

1999

Abstract

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 Z_n 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, Z_4, Z₂[x]/x²Z₂[x], or a four element field.

Comments

Previously linked to as: http://ccdl.libraries.claremont.edu/u?/irw,440.

Article can also be found at http://dml.cz/dmlcz/119099

Rights Information

© 1999 Charles University in Prague

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.

Recommended Citation

Henriksen, Melvin, and F. A. Smith. "The Bordalo order on a commutative ring." Commentationes Mathematicae Universitatis Carolinae 40.3 (1999): 429-440.