Warfield has recently defined a new class of invariants for mixed modules over a discrete valuation ring. These invariants, along with the Ulm invariants, enable Warfield to prove an analogue to Ulm's theorem. Warfield's definition contains two shortcomings. The invariants are defined for a limited class of modules. Moreover it is difficult to show that the invariants are well defined. This paper defines a new invariant which coincides with that of Warfield, and overcomes both difficulties.

No abstract

An experimental “management‐by‐objectives” performance system was operated by the Libraries and Information Systems Center of Bell Laboratories during 1973. It was found that, though the system was very effective for work planning and the development of people, difficulties were encountered in applying it to certain classes of employees, particularly those doing public service or routine work.

A new selection policy has recently been imple-prepared with the advice of users and supported by mented by the Bell Laboratories Library Network. It is several existing computer-aided systems. Interests are based on selection profiles for each of the member represented both by Dewey Decimal Classification libraries. These profiles, which detail the subjects in numbers and subject descriptors. The profiles provide a which books and other materials will be collected, are formal means by which management can bring available resources directly to bear on user needs.

Abstract. Warfield has defined the concept of a T*-module over a discrete valuation ring and has proved a classification theorem for these modules. In this paper, the invariant 5 defined by the author is extended. This allows a generalization of the classification theorem of Warfield.1. Preliminaries. R will be a discrete valuation ring throughout, and /» will represent a generator of its maximal ideal. The word "module" will mean R -module. Standard terminology of abelian groups will be used (see Fuchs [1], especially §79). The exception to this is that we will define the height of 0 to be oo', and assume a < co < oo' for all ordinals a. The indicator of an element x will be denoted H(x). An indicator is called proper if it does not contain oo'. If A is a submodule of M and a is an ordinal or co, the ath relative Ulm invariant will be denoted f(a, M, A).In [5], (announced in [4]), Warfield defined a T-module as a module that can be defined in terms of generators and relations in such a way that the only relations are of the form/»* = 0 orpx = y. A summand of a F-module is called a T*-module. A module M has torsion free rank one if, for any two elements x and y of infinite order in M, there are nonzero elements r and s in R such that rx = sy. Warfield has shown that if M is a F-module, it is either a torsion module, or a direct sum of modules of torsion free rank one.A subset X of a module M is a decomposition basis if [X] is the free module on X, M/[X] is torsion, and h(^r¡x¡) = min(rj(/-,x,)}, for r¡ E R, x¡ G X. (Here /i(x) denotes the height of x.) It has been shown by Warfield that a 7^-module M has a decomposition basis X such that [X] is nice in M.In §2, we define the concept of stability, which is stronger than niceness. (ii) every coset m + A of infinite order contains an element x such that /» 'x