Progress in Communicative Algebra

This is the first of two volumes of a state-of-the-art survey article collection which emanates from three commutative algebra sessions at an AMS meeting in 2009. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. The current trends in three of the most active areas of commutative algebra are presented: non-noetherian rings (factorization, ideal theory, integrality), advances from the homological study of noetherian rings (the local theory, graded situation and its interactions with combinatorics and geometry). This fist volume concentrates on combinatorics and homology.