On an n-fold torsion theory in the category rM

Abstract

In [4], Dickson has developed the torsion theory for an Abelian category. In this paper, generalizing the concept of torsion theory for the category RM of left R-modules, we shall define, for any integer n > 1, the concept of an n-fold torsion theory for RM and discuss its properties. A 2-fold torsion theory is the same thing as a torsion theory for RM as defined by Dickson [4] and a 3-fold torsion theory is nothing but a TTF-theory as defined by Jans [5]. We shall begin in Section 1 by reviewing the definitions and basic properties of a torsion theory and a TTF-theory. In Section 2, we shall give the definition of an n-fold torsion theory for RM and that of its length. A 3-fold torsion theory with length 2 is precisely a centrally splitting TTF-theory (see Bernhardt [3]). The main theorem of this section will give some characterizations of a centrally splitting TTFtheory (Theorem 2.7). We shall treat, in Section 3, 4-fold torsion theories, and give some necessary and sufficient conditions for any 4-fold torsion theory to have the length 2 (Proposition 3.1). As a consequence of this proposition, we can show that, for 71 > 4, any n-fold torsion theory has the length 2 (Theorem 3.3). We can also claim from this theorem that there exist only four different types of n-fold torsion theories (Theorem 3.4). Finally, in Section 4, we shall discuss, in particular, n-fold torsion theories for RM over a semisimple ring with minimum condition, over a commutative ring, and also over a semiprime ring. Examples will be given to show that each type of n-fold torsion theories exists. Throughout this paper, R will denote an associative ring with identity and RM the category of unital left R-modules and R-homomorphisms. We shall deal almost exclusively with left R-modules, and so unless otherwise