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Let \( G \) be a connected graph of order 3 or more and \( c : E(G) \to \mathbb{Z}_k \) (\( k \geq 2 \)) an edge coloring of \( G \) where adjacent edges may be colored the same. The color sum \( s(v) \) of a vertex \( v \) of \( G \) is the sum in \( \mathbb{Z}_k \) of the colors of the edges incident with \( v \). An edge coloring \( c \) is a modular neighbor-distinguishing \( k \)-edge coloring of \( G \) if \( s(u) \neq s(v) \) in \( \mathbb{Z}_k \) for all pairs \( u, v \) of adjacent vertices of \( G \). The modular chromatic index \( \chi_m'(G) \) of \( G \) is the minimum \( k \) for which \( G \) has a modular neighbor-distinguishing \( k \)-edge coloring. For every graph \( G \), it follows that \( \chi_m'(G) \geq \chi(G) \). In particular, it is shown that if \( G \) is a graph with \( \chi(G) \equiv 2 \mod 4 \) for which every proper \( \chi(G) \)-coloring of \( G \) results in color classes of odd size, then \( \chi_m'(G) > \chi(G) \). The modular chromatic indices of several well-known classes of graphs are determined. It is shown that if \( G \) is a connected bipartite graph, then \( 2 \leq \chi_m'(G) \leq 3 \) and it is determined when each of these two values occurs. There is a discussion on the relationship between \( \chi_m'(G) \) and \( \chi_m'(H) \) when \( H \) is a subgraph of \( G \).