Frank Okoh

Applications of linear functionals to Kronecker modules. II

This paper continues the investigation of a class of modules Er⩾ 2, constructed from submodules of the module of rational functions K(X), over the polynomial ring K[X], given by a height function h; sequences of linear functionals L=(l2,…,lr on K(X); and a sequence of positive integers (m2,…,mr. The emphasis is on when 1<card{θϵK ∪ {∞}:h(θ)≠}<∞. To each subset L′ of {l2,…,lr} is attached a field extension KL′ of the prime subfield of K. Let ML1 and ML2 be modules constructed from the sequences of linear functionals L1 and L2. If ML1 is isomorphic to ML2, then KL1 KL2 are contained in finitely generated field extensions of each other. Let Li=L ⧹ li, i=2,…,r. If ML is not purely simple, then for some i in { 2,…,r}, Kli is contained in a finitely generated extension of KLi. These results are then used to obtain the following theorems: assuming that 1<supp h<∞, then there are exactly ℵ0 isomorphism classes of completely decomposable modules in Er; there are at least c isomorphism classes of (i) indecomposable but not purely simple modules in Er, (ii) purely simple modules in Er (c = the cardinality of the continuum).

DIRECT SUM DECOMPOSITION OF THE PRODUCT OF PREINJECTIVE MODULES OVER RIGHT PURE SEMISIMPLE HEREDITARY RINGS

ABSTRACT In his recent work, [1] and [2], on the pure semisimplicity conjecture Simson raised two problems about the structure of the direct sum decomposition of the direct product modulo the direct sum of indecomposable preinjective modules over right pure semisimple hereditary rings. The main goal of this paper is the proof of a theorem that resolves one of these problems and provides a partial answer to the other.

DIRECT PRODUCTS OF MODULES AND THE PURE SEMISIMPLICITY CONJECTURE

It is shown that, if R is either an Artin algebra or a commutative noetherian domain of Krull dimension 1, then infinite direct products of R-modules resist direct sum decomposition as follows: If is a family of non-isomorphic, finitely generated, indecomposable R-modules, then is not a direct sum of finitely generated modules. The bearing of this direct product condition on the pure semisimplicity problem is discussed.

Rational functions and kronecker modules

There exists a function f: K → K, on any infinite field K, such that for any rational function r(X), f and r agree on a finite but not empty set. The purely simple modules of rank two over the Kronecker algebra may be all indexed by three parameters: a positive integer n, a height function and a K-linear map α: K(X) → K. When the support of h,i.e. { θ ∊ K: h(θ) ≥ 0}, has lesser cardinality than K,then the integer n is redundant. If the support of hhas the same cardinality as K, then for each there exists a purely simple, rank two, A-module E(n,h,α),not isomorphic to any other purely simple module of rank two, which is indexed by a positive integer less than n. The construction of this E(n,h,α) uses the function f.

Polynomial extensions of IDF-domains and of IDPF-domains

An integral domain is IDF if every non-zero element has only finitely many non-associate irreducible divisors. We investigate when R IDF implies that the ring of polynomials R[T] is IDF. This is true when R is Noetherian and integrally closed, in particular when R is the coordinate ring of a non-singular variety. Some coordinate rings R of singular varieties also give R[T] IDF. Analogous results for the related concept of IDPF are also given. The main result on IDF in this paper states that every countable domain embeds in another countable domain R such that R has no irreducible elements, hence vacuously IDF, and the polynomial ring R[T] is not IDF. This resolves an open question. It is also shown that some subrings R of the ring of Gaussian integers known to be IDPF also have the property that R[T] is not IDPF.

Separable modules over finite-dimensional algebras

In this paper all modules are unital right modules over finite-dimensional algebras over algebraically closed fields, K. A submodule L of a module N is said to be a pure submodule of N if L is a direct sumand of M whenever L t M c N and M/L is finite-dimensional. A module L is pure-injective if whenever L is pure in M, it is a direct summand of M. A module N is pure-projective if every exact sequence 0 4 L + M 4 N -+ 0 with L pure in M splits. The following proposition is easy to prove:

Bouquets of Baer modules

Abstract In this paper some transcendental numbers are used to construct infinite-dimensional indecomposable Baer modules. Let R be a ring whose category of modules has a torsion theory. An R -module, M , is Baer if every extension of M by any torsion R -module splits. In this paper, R will be a path algebra, i.e., an algebra whose basis over a field K are the vertices and paths of a directed graph. Multiplication is given by path composition. When R is a path algebra obtained from an extended Coxeter-Dynkin diagram with no oriented cycles, we characterize Baer modules of countable rank. This characterization is used to show that modules constructed from Liouville sequences yield a family, = { B n } ∞ n =0 , of Baer modules satisfying the following conditions: every extension of B m by B n splits for every pair ( m,n ); if m ≠ n , B m is not isomorphic to B n , while automorphisms of B n are given by multiplications by nonzero elements of K . Each B n is shown to be a submodule of a rank-one module. Another application of our characterization is the determination of the rank-one modules with the property that every submodule of infinite rank has a nonzero direct summand that is Baer. In analogy with ℵ r -free modules, we define ℵ r -Baer modules and give an example of an ℵ 1 -Baer module that is not Baer. The existence of a Baer module, M , that is not a direct sum of Baer modules of countable rank is also proved. However every nonzero submodule of M has a nonzero direct summand. A problem suggested by these results is the existence and structure of indecomposable Baer modules of uncountable rank.

Direct sums and direct products of finite-dimensional modules over path algebras

Abstract The algebras in this paper are over the associative algebras R obtained from extended Coxeter-Dynkin quivers with no oriented cycles. The finite-dimensional indecomposable R -modules can, in principle, be described. Taking direct products and direct sums, respectively, of finite-dimensional R -modules over an infinite indexing set are two natural ways of getting infinite-dimensional R -modules. The latter are the infinite-dimensional pure-projective modules and direct summands of the former are the pure-injective modules. The focus in this paper is on these two classes of infinite-dimensional modules. Every module is a submodule of a pure-injective module and a quotient of a pure-projective module. When is an extension of a pure-injective module by a pure-injective module pure-injective? This question is answered in this paper. The answer is analogous to the answer of the corresponding question for pure-projective modules. The structures of some quotients of direct products of finite-dimensional modules are also obtained.

On Fragility of Generalizations of Factoriality

Any class of domains, in particular a class of domains that arises from generalizations of factoriality, invites questions about its stability under the standard operations. One of these generalizations of factoriality is the one that requires that every nonzero element be contained in only finitely many principal prime ideals of height one. We use this property to settle all the open cases in the literature on stability of generalizations of factoriality under the standard ring extensions. The paper provides a compendium on the stability, under ring extensions, of all the known generalizations of factoriality. We also use stability properties of factorization in extensions of valuation domains to give a new characterization of discrete valuation domains.

Curves arising from Kronecker modules

Abstract Let K be an algebraically closed field. A Kronecker module M is a pair of K -vector spaces ( S , T ) together with a K -bilinear map K 2 × S → T . The space S is called the domain space of M , while T is called the range space of M . To each power series α in K [[ X ]] we attach a Kronecker module P α whose domain and range spaces are denoted by V − and V , respectively. Both V − and V are modules over the endomorphism algebra End P α of P α . We show that if End P α is non-trivial, then the sequence of coefficients of α is defined by a linear or a quadratic recursion. In the quadratic case End P α is the coordinate ring of an affine curve. An affine curve is called realizable when its coordinate ring is isomorphic to some End P α . We show that the realizable curves can be constructed, up to birational equivalence, by pairs of non-zero polynomials ( p , q ) with deg q p . The curves realized herein are embedded in K d where d =deg p . The planarity of these curves when d ⩾3 remains open. Our main result on realization of curves is that a cubic curve is realizable if and only if its coordinate ring is either K [ X ] or the coordinate ring of a cubic in Weierstrass form.