A note on the number of distinct elements and zero-sum subsequence lengths in cyclic groups

In this short note we investigate zero-sum sequences in finite abelian groups, examining the relationship between the sequence's support size, that is the number of distinct elements, and its properties concerning zero-sums. In particular, for sequences $S$ in a cyclic group, we establish a direct connection between $MZ(S)$, the length of the shortest nonempty subsequence summing to zero and the number of distinct values in $S$. Our results reveal that sequences with larger support must contain shorter non-empty zero-sum subsequences, in line with classical zero-sum results. Additionally, we present one application of our main result to a factorization of ideals problem in rings of integers of a number field.