Geodesic
The Bordalo order on a commutative ring
Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) no. 3, pp. 429-440.

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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.
Classification : 03G10, 06A06, 06F25, 11A07, 13A99
Keywords: commutative ring; reduced ring; integral domain; field; connected ring; \linebreak Boolean ring; weak Baer Ring; regular element; annihilator; nilpotents; idempotents; cover; partial order; incomparable elements; lattice; modular lattice; distributive lattice 
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Henriksen, Melvin; Smith, F. A. The Bordalo order on a commutative ring. Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) no. 3, pp. 429-440. http://geodesic.mathdoc.fr/item/CMUC_1999__40_3_a2/