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Extensions of modules characterized by finite sequences of linear functionals. (English) Zbl 0804.16032

Summary: Let \(S\) be an algebra over an algebraically closed field, \(K\). If \(S\) is different from \(K\), then it contains \(K^ 2 = K \oplus K\) as a \(K\)- vector subspace, e.g., \(S = K[\zeta]\), the polynomial ring in one variable over \(K\). Then any \(S\)-module \(M\) gives rise to a pair of \(K\)- vector spaces \({\mathbf M} = (M,M)\) and a \(K\)-bilinear map from \(K^ 2 \times M\) to \(M\). This makes \(M\) a right module over the matrix ring, \(R = \bigl[ {K \;K^ 2\atop 0 \;K}\bigr]\). An \(R\)-module isomorphic to \({\mathbf M} = (M,M)\) where \(M\) is a \(K[\zeta]\)-module is said to be nonsingular; an \(R\)-module is torsionfree if it is isomorphic to a submodule of \({\mathbf M} = (M,M)\) where \(M\) is a torsion-free \(K[\zeta]\)- module. In this paper it is shown that extensions \(X\) of finite- dimensional torsion-free \(R\)-modules \(U\) by nonsingular \(R\)-modules are characterized by finite sequences of linear functionals. This provides an upper bound on the dimension of the vector space of extensions of \(U\) by \(V\). Questions about such extensions become questions on the existence of linear functionals with appropriate properties. In particular,when \(V = (K(\zeta),K(\zeta))\), where \(K(\zeta)\) is the \(K[\zeta]\)-module of rational functions the setup provides a fertile source of indecomposable infinite-dimensional \(R\)-modules. We describe extensions, \(X\), of \(U\) by \(V\), with the property that the endomorphism ring of \(X\) is an integral domain. Moreover, \(X\) shares an infinite-dimensional indecomposable submodule with \(V\).

MSC:

16S70 Extensions of associative rings by ideals
16S50 Endomorphism rings; matrix rings
15A21 Canonical forms, reductions, classification
16D80 Other classes of modules and ideals in associative algebras
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
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