Hereditary Classes Research Articles

On the Ring of Quotients at a Prime Ideal of a Right Noetherian Ring

J. Lambek and G. Michler [3] have initiated the study of a ring of quotients RP associated with a two-sided prime ideal P in a right noetherian ring R. The ring RP is the quotient ring (in the sense of [1]) associated with the hereditary torsion class τ consisting of all right R-modules M for which HomR(M, ER(R/P)) = 0, where ER(X) is the injective hull of the R-module X.In the present paper, we shall study further the properties of the ring RP. The main results are Theorems 4.3 and 4.6. Theorem 4.3 gives necessary and sufficient conditions for the torsion class associated with P to have property (T), as well as some properties of RP when these conditions are indeed satisfied, while Theorem 4.6 gives necessary and sufficient conditions for R to satisfy the right Ore condition with respect to (P).

Rings whose modules form few torsion classes

Characterizations are obtained of rings R such that the only torsion classes (respectively, hereditary torsion classes) of left unital R-modules are {0} and the class of all modules.

Strongly hereditary radicals

A radical class (P of some universal class SW of not necessarily associative rings is hereditary if RCG implies IC'( for all ideals I of R, and will be called strongly hereditary (terminology suggested by M. Slater) if for all R CW we have G(I) = InvP(R) for all ideals I of R. (Note that in [2 ] it is the classes with this last property which are called hereditary.) It is well known that these properties are equivalent if SW is any universal class of associative or alternative rings [1, Lemma 1, p. 595 together with Corollary 1, p. 597 for associative rings or Corollary 1, p. 602 for alternative rings]. It is also clear that every strongly hereditary class is hereditary, but the converse is not true in general. This was shown by E. P. Armendariz in [2] using either of the rings of [3, p. 1116]. Note that by [1, Lemma 2, p. 595] hereditary radical classes are strongly hereditary if their semisimple classes are also hereditary. Since the hereditary and strongly hereditary properties are in general distinct, E. P. Armendariz has raised the question (private communication) of the existence of a strongly hereditary radical construction analogous to the lower radical. The key to such a construction is provided by the following alternative criterion for a strongly hereditary radical:

Social-Economic Classes in Soviet Russia

T HE STRUCTURE of social-economic classes in Soviet Russia has been patterned by the institutions of the revolution. It has been built upon a living web of traditional attitudes and social values which come from Tsarist times, however, and it has been shaped in its details by the external pressures of imminent and actual invasion and war. The Russia of today, consequently, is not the result of any given blueprint for an ideal society; the Russians themselves would be the last to claim anything of the kind. Like other peoples, Soviet Russia has had to form her institutions and her ways of life out of the materials and the circumstances that were available to her. In her case, that has meant a people whose masses were largely illiterate and slow of mind and body a quarter of a century ago. It has meant a traditional respect for, even awe of, authority that had been imbibed for centuries from family life, church and state. Even with new life and new goals, it has meant the threat of war afid invasion from without, as well as of counter-revolution from within, for a large part of the twentyfive years which span the life of the Soviets to date. To the sociologist, however, a large measure of interest springs from the fact that in part, at least, the organization of modern Russia stems from a deliberate plan. Particularly the class structure has been influenced, even formed, by a series of events and a set of institutions which have arisen from a consciously made program. The program, certainly, has been formed to fit a given people in a given set of circumstances. The costs of its production in both economic and social terms have been high. The stakes, likewise, have been high, however, and the alternatives to the game as played by the Russian Communist leadership have not been inviting. Economic imperialism from without, and the hereditary class divergence from within offered a wide variety of exploitative opportunities that boded ill for a backward, ignorant people, in i9i8. After nearly twenty-five years of the Soviet government and over ten years of planned economy, therefore, one may well take stock of the results. The writer recognizes that in an article of this sort and at this time a large measure of apparently subjective observation must be the basis for the analysis which is presented here and the conclusions that are reached; the limitations of this article will not permit the inclusion of detailed evidence. Fifteen years of careful observation and gathering of data, however, and the accumulation of a mounting volume of general information regarding Soviet life and patterns of thought, which are doubtless available to the readers of this journal, are excuse enough for undertaking an evaluation of this sort. One may not, also, predict the influences of invasion and war, such as the Russians have suffered, in altering any set of institutions among any people. Of necessity, one must discuss the structure of Soviet Russia before the present war, predominantly. It would seem to the writer, nevertheless, that Soviet patterns were clearly enough defined, before the war, and the success of the war as waged by the Soviets is assured enough to allow us to expect a considerable measure of internal likeness between the social structure of the Russian people before and after the present conflict. It is the backward nation of 19I7, revolutionized and reconstructed in turn by the Bolsheviki and Communist leadership during the nineteen twenties, and nationalized and crystallized by the threat of war and the arming of the Russian people to meet that war, in the nineteen thirties, which has made the Soviet Union of I94I-