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By R. Göbel, C. Metelli, A. Orsatti, L. Salce

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3. w C • o. B(n) 2 p -bounded countable valuated group A G of length that is not a direct sum of cyclics. order p be of order x1 Let 2 shall n th and value and value sequence . 0 The p • 1 • 2 • 3 • 2 • 3 • 4 Ulm invariant of show that G (n-l,n). G is has no finite for 0. For Let n '1- 1, n G > let x n have be the subgroup of and summand with a 1 G(p) c G(2). nonzero 0 th We Ulm Subgroups of Bounded Abelian Groups invariant. A Suppose that is nonzero. Let represents the Oth that if vx A so G, a 0, y '" x - uy n· upy) > B, and that the be the projection on then there is y Now Then A.

14(1973) 833-837. , /~finite Abelian Groups, New York, 1970. Hunter, R. and E. Walker, Valuated p-Groups, Lecture Notes in Mathematics 874(1981) 350-373. , F. Richman and E. A. Walker, Finite direct sums of Cyclic Valuated p-Groups, Pacific. J. Hath. 69(1977) 97-104. Richman and E. A. Walker, Simply Presented Valuated Abelian p-Groups, J. Of Algebra, 49(1977) 125-133. , and E. A. Walker, Valuated Groups, J. of Algebra, 56(1979) 145-167. w A COMBINATORIAL THEOREM AND ENDOMORPHISM RINGS OF ABELIAN GROUPS II Saharon Shelah (*) University of Jerusalem §0 Introduction This paper was originally part of [Sh 8] • It was separated technical reasons and partly extendend, particularly in §§5,6.

3. , be fixed (with P'Y\ 6 each J'" be { : a) predicate L

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