Trivial extensions defined by coherent-like conditions

This paper investigates coherent-like conditions and related properties that a trivial extension R := A alpha E might inherit from the ring A for some classes of modules E. It captures previous results dealing primarily with coherence, and also establishes satisfactory analogues of well-known coherence-like results on pullback constructions. Our results generate new families of examples of rings (with zerodivisors) subject to a given coherent-like condition.