String rewriting reductions of the form $t\to_R^+ utv$, called
loops, are the most frequent cause of infinite reductions
(non- termination). Regarded as a model of computation, infinite
reductions are unwanted whence their static detection is important.
There are string rewriting systems which admit infinite reductions
although they admit no loops. Their non-termination is particularly
difficult to uncover. We present a few necessary conditions
for the existence of loops, and thus establish a means to recognize
the difficult case. We show in detail four relevant criteria:
(i) the existence of loops is characterized by the existence of
looping forward closures; (ii) dummy elimination, a
non-termination preserving transformation method, also preserves the
existence of loops; (iii) dummy introduction, a
transformation method that supports subsequent dummy elimination,
likewise preserves loops; (iv) bordered systems can be
reduced to smaller systems on a larger alphabet, preserving the
existence and the non-existence of loops. We illustrate the power of
the four methods by giving a two-rule string rewriting system
over a two-letter alphabet which admits an infinite reduction
but no loop. So far, the least known such system had three rules.