  We present a new test for studying asphericity and diagrammatic reducibility
of group presentations. Our test can be applied to prove diagrammatic
reducibility in cases where the classical weight test fails. We use this
criterion to generalize results of J. Howie and S.M. Gersten on asphericity of
LOTs and of Adian presentations, and derive new results on solvability of
equations over groups. We also use our methods to investigate a conjecture of
S.V. Ivanov related to Kaplansky's problem on zero divisors: we strengthen
Ivanov's result for locally indicable groups and prove a weak version of the
conjecture.
