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If $R$ is a commutative ring with identity and $\le$ is defined by letting $a\le b$ mean $ab=a$ or $a=b$, then $(R,\le )$ is a partially ordered ring. Necessary and sufficient conditions on $R$ are given for $(R,\le )$ 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 $n\ge 2$. In particular, if $R$ is reduced, then $(R,\le )$ is a lattice iff $R$ is a weak Baer ring, and $(R,\le )$ is a distributive lattice iff $R$ is a Boolean ring, $Z_3,Z_4$, $Z_2\left[x\right]/x^2{Z}_2\left[x\right]$, or a four element field.

In 1950 in volume 1 of Proc. Amer. Math. Soc., B. Brown and N. McCoy showed that every (not necessarily commutative) ring $R$ has an ideal $𝔐\left(R\right)$ consisting of elements $a$ for which there is an $x$ such that $axa=a$, and maximal with respect to this property. Considering only the case when $R$ is commutative and has an identity element, it is often not easy to determine when $𝔐\left(R\right)$ is not just the zero ideal. We determine when this happens in a number of cases: Namely when at least one of $a$ or $1-a$ has a von Neumann inverse,...

A lattice-ordered ring $ℝ$ is called an if each of its order ideals is a ring ideal. Generalizing earlier work of Basly and Triki, OIRI-rings are characterized as those $f$-rings $ℝ$ such that $ℝ/𝕀$ is contained in an $f$-ring with an identity element that is a strong order unit for some nil $l$-ideal $𝕀$ of $ℝ$. In particular, if $P\left(ℝ\right)$ denotes the set of nilpotent elements of the $f$-ring $ℝ$, then $ℝ$ is an OIRI-ring if and only if $ℝ/P\left(ℝ\right)$ is contained in an $f$-ring with an identity element that is a strong order unit.