Exactness, Tor and Flat Modules Over

In this paper, we principally explore flat modules over a commutative ring with identity. We do this in relation to projective and injective modules with the help of derived functors like Tor and Ext. We also consider an extension of the property of flatness and induce analogies with the “special cases” occurring in flat modules. We obtain some results on flatness in the context of a noetherian ring. We also characterize flat modules generated by one element and obtain a necessary condition for flatness of finitely generated modules. I. DEFINITIONS • Flat modules: Those modules F for which the functor _⊗F is exact are termed flat modules. Let R be a commutative ring with identity and consider the modules over R. • Derived functors: Let T be an additive functor and N be an R-module. If the Ci’s are all projective modules and the following sequence is exact: .... →...→C1→C0→N→0, then we have a projective resolution of N. The iderived functor with respect to T is the homology module at T(Ci ), i.e., the quotient of the kernel of the map leaving T(Ci) to the image of the map entering it from T(Ci-1). It can be proved that the derived functor is determined uniquely up to isomorphism by any projective resolution [1-7]. • Exact sequence: A sequence of maps A→B→C is said to be exact at B if the image of the map “entering” B is equal to the kernel of the map “leaving” B. • Functor: It is a map from the set of Rmodules to itself. • Exact functor: A functor T which when applied to all the terms of an exact sequence induces another exact sequence is said to be exact. The functor T in question should be additive, i.e. if f is a map from M to N, there should exist an induced map T(f) from T(M) to T(N). • Projective modules: A module P having the following property: If p is a map from M onto N and f is a map from P to N, there exists a map g from P to M such that pog = f. In other words, P is such that the functor Hom(P,_) is exact. II. FLATNESS Let R be a commutative ring with identity [2]. Flatness may be defined through either of the following equivalent conditions: • Injective modules: A module Q having the following property: If i is a one to one map from K to M and f is a map from K to Q, there exists a map g from M to Q such that goI = f. In other words, the functor Hom(_,Q) is exact. (i) If P→Q is a monomorphism of Rmodules, the induced map from M⊗P → M⊗Q is also a monomorphism. (ii) The functor _ ⊗M is exact.