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This paper continues the study of codivisible modules, whose definition is a “dualization” of Lambek's concept [4] of a divisible module relative to a torsion theory. The main purpose of this work is to give a solution to the following problem posed by Bland [2]: “It would be interesting to know under what conditions the universal existence of codivisible covers implies that of projective covers.” The if-and-only-if nature of the solution, which is given in Theorems 2 and 4, shows that our sufficient conditions are “best possible” conditions. The method of the solution introduces the concept of a pseudo-hereditary torsion theory, which may be of interest in its own right; in particular, every hereditary torsion theory and every faithful torsion theory is pseudo-hereditary. The main results for a pseudo-hereditary torsion theory (T, F) relate the conditions, R/T(R) is semiperfect and R/T(R) is left perfect, to the existence of codivisible covers (see Theorems 1 and 3 and Corollaries 1 and 2).

A prominent question in the study of modules over an integral domain has been: “When is the torsion submodule t(A) of a module A a direct summand of A?” A module is said to split when its torsion module is a direct summand. Clearly, every cyclic module over an integral domain splits. Interesting splitting problems have been explored by Kaplansky [14; 15], Rotman [20], Chase [4], and others.Recently, many concepts of torsion have been proposed for modules over arbitrary associative rings with identity. Two of the most important of these concepts are Goldie's torsion theory (see [1; 12; 22]) and the simple torsion theory (see [5; 6; 8; 9; 23], and their references).

Introduction. Let R be an associative ring with identity and let RAN (respectively 9ifR) denote the category of unitary left (respectively right) R-modules. Dickson [4] has given an axiomatic treatment of torsion for abelian categories. Specializing his definition, a torsion theory for RA is defined to be a pair (3, 5) of classes of left R-modules such that (a) 3n5CM=O0}, (b) 3 is closed under homomorphic images, (c) 5f is closed under submodules, (d) for every left R-module M there exists a submodule T(M) of M with T(M) Ez and M/T(M) E . Throughout this paper two additional properties are required of a torsion theory (e) RREf, (f) 5 is closed under submodules. A torsion theory for OJMR is similarly defined. If (r, 5) is a torsion theory those modules in 3 are said to be torsion and those in 5f are said to be torsion-free. Specific examples of torsion theories having these properties are the usual torsion theory for abelian groups, the torsion theory for left R-modules over a left Ore ring studied by Levy [9] and the E(R)-torsion theory considered by Jans [7].

A brief review of Saint-Venant torsion theory is given and the advantages and disadvantages of the various forms in which the equations can be expressed are discussed in relation to electrical analogue computation. It is shown how, by the use of a simple passive resistance network, solutions can easily be obtained both for simply connected regions as well as for the analytically difficult problem of multiply connected regions. Examples of the application of the electrical-analogue method to the problems of plastic torsion, the torsion of compound bars, thick hollow sections and thin-walled sections filled with a dissimilar material, are given.

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

Chase has given several characterizations of a right coherent ring, among which are: every direct product of copies of the ring is left-flat; and every finitely generated submodule of a free right module is finitely related. We extend his results to obtain conditions for the ring of quotients of a ring with respect to a torsion theory to be coherent.

A torsion theory partitions the spectrum of the base ring into two sets. Over a Noetherian ring, every suitable partition of the spectrum gives rise to one and only one torsion theory. It is possible to know whether a module is torsion or torsion-free by looking at its associated primes. The example of the polynomial torsion theory is developed. Notations and terminology. All rings are commutative with 1. We denote by AssA(M) the set of weakly associated primes of an A-module M. A prime p is in ASSA(M) if p is minimal among the primes containing the annihilator annA(x) of a nonzero element x of M. We denote by AssA(M) the set of strongly associated primes of an Amodule M. A prime p is in AssA(M) if it is the annihilator annA(x) of a nonzero element x of M. We denote by SuppA(M) the support of an A-module M. A prime p is in SUPPA(M) if the localization MJ is not null. We send the reader to Lambek [5] for the definition and immediate properties of a torsion theory. We denote by T(M) the torsion submodule of a module M. 1. Partition of the spectrum. Throughout this section A is a ring with a torsion theory. PROPOSITION 1. 1. Let M be be an A-module. (1) If A/p is not a torsion module, for every p in ASSA(M), then M is a torsion-free module. (2) If M is a torsion-free module, then A/p is torsion-free for every p in AssA(M). PROOF. (1) Suppose that M is not a torsion-free module. The torsion submodule T(M) must contain a nonzero element x and AIannA(x)-Ax, Presented to the Society, May 1, 1972; received by the editors May 5, 1972. AMS (MOS) subject classifications (1970). Primary 13C10.

Bending rigidity of a crankcase of high speed internal combustion engines is usually smaller in the horizontal-transverse direction of the engine, because of its common hanger-bearing-type construction. For this reason, the elastic vibration appears mainly in the horizontal direction. In our research the bending deflection of a crankcase was measured under such conditions that the load was applied to the middle point of a crankcase in the horizontal-transverse direction being simply supported at the both end-bearing centres. In analysing the results of bending tests with several models of crankcase, the calculating equations for the bending deflection of a crankcase were induced from "Torsion Theory of Thin Sections"with some modifications. The results of bending tests on regular engines show that those equations are practically adequate to calculate the deflection of a crankcase.

A ring R is called, according to [2], a left p.p. ring if any principal left ideal of R is projective. A ring which is left and right p.p. is called a p.p. ring.In this short note we shall give some additional remarks to A. Hattori [2]. In Proposition 1 we shall give a characterization of commutative p.p. rings, and in Proposition 3 we shall give a generalization of Proposition 17 and 18 in [2], which shows also that the modified torsion theory over commutative p.p. rings coincides with the usual torsion theory.

In this paper, the following condition for a torsion theory in the sense of S. E. Dickson is examined: ( P \text {P} ) Every nonzero torsionfree module contains a nonzero projective submodule. A special relationship between condition ( P \text {P} ) and Goldie’s torsion theory is shown; and the rings, for which every nontrivial torsion theory satisfies condition ( P \text {P} ), are classified.

Since the TTF-theory was initiated by Jams [~], various authors have made interesting studies of the theory, particularly, over semi-perfect rings. The present paper is to establish several basic properties of TTF-classes, which provide a generalization and refinement of their results. It will be shown, among others, that if (~, ~) is a torsion theory over a ring R on the left with TTF-class ~ determined by an idem~otent ideal I, then d7 is a TTF-class or hereditary if and only if I is a direct s,~d of R or ~I is flat as a right R-module respectively. Let R be a ring with unit element. A class of (unital) left R-modules is called a torsion class if it is closed under homomorphic images, direct sums, and group extensions, while it is called a torsion-free class if it is closed under submodules, direct products, and group extensions. Let C be any class of left R-modules. Define C t to be the class of those left R-modules X for which HOmR(X,M ) = 0 for all M in ~. Then ~ is a torsion class. Similarly, if we define ~r to be the class of those left R-modules Y for which HomR(M,Y ) = 0 for all M in ~, then O r is a torsion-free class. Conversely, it was shown by Dickson [3] that every torsion class and every torslon-free class is given in this way; more precisely, ~ is a torsion class if and only if (~r) 9 = ~, while ~ is a torsion-free class if and only if (d~) r = ~. A torsion class is called hereditary if it is closed under submodules. A necessary and sufficient condition for this is that the associated torsion-free class ~r is closed under inJective envelopes, as was proved also by Dickson. Now, ~ is called

When we consider modules A over a ring R which is not a commutative integral domain, the usual torsion theory becomes somewhat inadequate, since zero-divisors of R are disregarded and since the torsion elements of A do not in general form a submodule. In this paper we shall try to remedy such defects by modifying the fundamental notions such as torsion modules, divisible modules, etc.