  Motivated by the recent definition of $AM$-property in locally solid vector
lattices [O. Zabeti, arXiv: 1912.00141v2 [math.FA]], in this note, we try to
investigate those results in the category of all locally solid lattice rings.
In fact, we characterize locally solid lattice rings in which order bounded
sets and bounded sets agree. Furthermore, with the aid of $AM$-property, we
find conditions under that, order bounded group homomorphisms and different
types of bounded group homomorphisms coincide. Moreover, we show that each
class of bounded order bounded group homomorphisms on a locally solid lattice
ring $X$ has the Lebegsue or the Levi property if and only if so is $X$.
