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A modular property of term rewriting systems is one that holds for the direct sum of two disjoint term rewriting systems, iff it holds for every involved term rewriting system. A term rewriting system is r-consistent, iff there is no term that can be rewritten to two different variables. We show that the subclass of left-linear and r-consistent term rewriting systems has the modular termination property. This subclass may also contain nonconfluent term rewriting systems. Since confluence implies r-consistency, this constitutes a generalisation of the theorem of Toyama, Klop, and Barendregt on the modularity of termination for confluent and left-linear term rewriting systems.

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