“…Recently, reductions of modules were introduced and developed in [16], and their "dual" concept was investigated in [20]. Also, there have been several recent papers in which a number of important theorems for ideals in Noetherian rings have been extended to Noetherian filtrations (for example, see [1, 8,9,14,15,18]). (Filtrations are generalizations of the sequence of powers of a given ideal, and there are many important filtrations (such as the sequence {q^} n >Q of symbolic powers of a primary ideal q and the sequence { (I n ) a } n >o of integral closures of the powers of an ideal /) which are generally not powers of an ideal, but which are quite often Noetherian filtrations.…”