Summary: Let \(G\) be a graph on \(n\) vertices. An irregular assignment of \(G\) is a weighting \(w: E(G)\to\{1,\ldots,m\}\) of the edge-set of \(G\) such that all weighted degrees \(w(v)=\sum_{v\in e}w(e)\) are distinct. The minimal number \(m\) for which this is possible is called the irregularity strength \(s(G)\) of \(G\). Lehel and others have shown that \(s(G)<\infty\) implies \(s(G)\leqq n-1\) for connected graphs on \(n\geq4\) vertices, and \(s(G)\leq 2n-3\) for arbitrary graphs. By using decompositions for the additive group \(\mathbb{Z}_ r\) (integers \(\mod r\)), these results are strengthened. Main Theorem: \(s(G)\leq n+1\) for any graph with \(s(G)<\infty\).