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 $n\geq2$. 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.
@article{119099,
author = {Melvin Henriksen and Frank A. Smith},
title = {The Bordalo order on a commutative ring},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {40},
year = {1999},
pages = {429-440},
zbl = {1011.06019},
mrnumber = {1732492},
language = {en},
url = {http://dml.mathdoc.fr/item/119099}
}
Henriksen, Melvin; Smith, Frank A. The Bordalo order on a commutative ring. Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) pp. 429-440. http://gdmltest.u-ga.fr/item/119099/
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