Homological Characterization of Rings: The Commutative Case

A large number of finiteness properties of commutative rings have ho-mological characterizations. For example, it is well known that for a ring to be Noetherian a condition most commonly described by the finite gen-eration of the ideals of the ring, it is necessary and su ¢ cient that arbitrary direct sums of injective modules be injective modules. One might speculate that this is the reason why homological algebra approaches in Noetherian settings yield such deep and beautiful results. The same phenomena can be observed in another large class of rings, the class of coherent rings. Chase (1960) attempted to answer the homological question: for what rings arbitrary direct products of at modules are at modules. The answer is that this holds true precisely when the ring is coherent. Chase provides no less then seven equivalent characterizations of this homological condition. The most well known are the two equivalent niteness conditions below: A ring R is called a coherent ring if every finitely generated ideal of