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.

It is well known that to every Boolean ring $ℛ$ can be assigned a Boolean algebra $ℬ$ whose operations are term operations of $ℛ$. Then a symmetric difference of $ℬ$ together with the meet operation recover the original ring operations of $ℛ$. The aim of this paper is to show for what a ring $ℛ$ a similar construction is possible. Of course, we do not construct a Boolean algebra but only so-called lattice-like structure which was introduced and treated by the authors in a previous paper. In particular, we reached...