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By David Arnold

The topic of this booklet is an exposition of connections among representations of finite in part ordered units and abelian teams. Emphasis is put all through on type, an outline of the gadgets as much as isomorphism, and computation of illustration variety, a degree of whilst category is possible. David M. Arnold is the Ralph and Jean hurricane Professor of arithmetic at Baylor collage. he's the writer of "Finite Rank Torsion unfastened Abelian teams and earrings" released within the Springer-Verlag Lecture Notes in arithmetic sequence, a co-editor for 2 volumes of convention lawsuits, and the writer of diverse articles in mathematical examine journals.

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The endomorphism ring of N E rep(S, k < x, is denoted by Endk(x ,y)N. Write EndkN for the endomorphism ring of N regarded as a k-representation, albeit one with possibly infinite dimension . o y» 32 1. y)N = EndkN. The category rep(S ,k) has fully wild representation type if FN is a fully faithful functor for some N . If rep(S, k) has fully wild representation type, then rep(S, k) has wild representation type, since a fully faithful functor preserves indecomposables and reflects isomorphisms. 1 The category rep(S5, k) has fully wild representation type.

Let k(x , y) denote the k-algebra of polynomials in noncommuting variables x and y with coefficients in a field k, and let modk(x, y) denote the category of k(x, y)-modules that have finite k-dimension. The endomorphism ring of N E rep(S, k < x, is denoted by Endk(x ,y)N. Write EndkN for the endomorphism ring of N regarded as a k-representation, albeit one with possibly infinite dimension . o y» 32 1. y)N = EndkN. The category rep(S ,k) has fully wild representation type if FN is a fully faithful functor for some N .

Hence , Vo = kx E9 Uo with kx £ Vi for i ::: t . Then Vi = Vi n Vo = kx E9 (Uo n Vj ) = kx E9 U, for each i :::t and Vi = Uo n Vj = U, for each i < t. This shows that ° V = (Vo, VI ," " Vm) = (kx, 0, ... ,0, kx, . . , kx) E9 (Uo , UI, . . , Um) = WE9U is a direct sum of representations, as desired . For the general case, write X as a direct sum of n indecomposable representations in rep(S , k). Use the above argument on an indecomposable summand of X and induction on n to see that U is a summand of V.

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