1. Introduction. Throughout k will denote a field. If a group T acts on a set A we say an element is T-orbital if its orbit is finite and write A r (A) for the subset of such elements. Let / be an ideal of a group algebra kA; we denote by 7 + the normal subgroup (/ + 1) D A of A. A subgroup B of an abelian torsion-free group A is said to be dense in A if A/B is a torsion-group. Let / be an ideal of a commutative ring K; then the spectrum Sp(/) of / is the set of all prime ideals P of K such that / < ? . If R is a ring, M is an -module and x e M we denote by s& R (x) the annihilator of x in R. We recall that a group T is said to have finite torsion-free rank if it has a finite series in which each factor is either infinite cyclic or locally finite; its torsion-free rank r o (F) is then defined to be the number of infinite cyclic factors in such a series.Let A be an abelian torsion-free group of finite rank acted upon by a group T and let / be an ideal of kA. The subgroup S r (A) of T of elements y such that / D kB = F D kB for some finitely generated dense subgroup B of A is said to be the standardiser of /. We will say that an ideal / of kA is locally prime if / D kB is a prime ideal of kB for some dense finitely generated subgroup B of A. It easily follows from Wilson's version [18, Section 3.11] of an important theorem of Brookes [1, Theorem A] that if A r (y4) = 1, / is a locally prime ideal of kA and 5 r (7) = T, then / + ^ 1. But, of course, 7 + may contain no non-trivial F-invariant subgroup.Let G be a group with a torsion-free abelian normal subgroup A of finite rank. In [12, Theorem E] Nabney proved that if M is any fcG-module which is not kA -torsion-free then there is an element a e A/\{0} such that akG = {akS)® kS kG, where S = S C (P) for some P G Sp (s& kA (a)). But, generally, if G has finite torsion-free rank then r o (S/C s (akS)) may be the same as r o (G) for any a e M\{0}. However, it would be very useful to find such a subgroup H of G that akG = (akH)

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Within the framework of the thermally activated process of the flux line or flux line bundles, and by time integration of the 1D equation of motion of the circulating current density J (ρ, t), which is suitable for thin superconducting films (R d, λ), we present numerical calculations of the current profiles, magnetization hysteresis loops and ac susceptibility χ n = χ n + iχ n for n = 1, 3 and 5 of a thin disc immersed in an axial time-dependent external magnetic field B a (t) = B dc + B ac cos(2πνt). Our calculated results are compared with those of the critical state model (CSM) and found to prove the approximate validity of the CSM below the irreversibility field. The differences between our computed results and those of the CSM are also discussed.

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The present paper is devoted to the study of finitely generated modules over nilpotent groups of finite rank and some problems connected with them. In it the approach suggested in [I] to the investigation of modules of this kind is developed. The essence of this approach is the local application of the methods and results of Hall [2, 3] concerning finitely generated solvable groups.Let A be a finitely generated ~G-module, ~ being a principal ideal ring, ~ being a locally almost polycyclic group of finite free rank. We shall denote by ~(~# the complete set of (nonassociated) As a rule, as the group ~ we consider a nilpotent group of finite free rank, and as either the ring Z , or ~<~> , the group algebra of the infinite cyclic group <~ over a finite field~. As is known, these two kinds of rings are the most important for applications.