  We propose a simple approach to formal deformations of associative algebras.
It exploits the machinery of multiplicative coresolutions of an associative
algebra A in the category of A-bimodules. Specifically, we show that certain
first-order deformations of A extend to all orders and we derive explicit
recurrent formulas determining this extension. In physical terms, this may be
regarded as the deformation quantization of noncommutative Poisson structures
on A.
