A monoid \(G\) is a `unique product monoid' (up-monoid) if for any two nonempty finite subsets \(A\) and \(B\) of \(G\), there is at least one \(c\in G\) which has a unique representation as \(c=ab\) with \(a\in A\) and \(b\in B\). Here the authors study the relationships between nil-related properties of a ring and those of the monoid ring \(RG\) where \(G\) is a up-monoid. Direct proofs are given of the (known) results that a ring is semiprime (resp. prime, reduced, a domain) if and only if \(RG\) has the corresponding property. The lower nilradicals of monoid rings are also computed. Many examples of up-groups are given to which the results can be applied.