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1. CJM 2014 (vol 67 pp. 286)

Bell, Jason P.; Lagarias, Jeffrey C.
 A Skolem-Mahler-Lech Theorem for Iterated Automorphisms of $K$-algebras This paper proves a commutative algebraic extension of a generalized Skolem-Mahler-Lech theorem due to the first author. Let $A$ be a finitely generated commutative $K$-algebra over a field of characteristic $0$, and let $\sigma$ be a $K$-algebra automorphism of $A$. Given ideals $I$ and $J$ of $A$, we show that the set $S$ of integers $m$ such that $\sigma^m(I) \supseteq J$ is a finite union of complete doubly infinite arithmetic progressions in $m$, up to the addition of a finite set. Alternatively, this result states that for an affine scheme $X$ of finite type over $K$, an automorphism $\sigma \in \operatorname{Aut}_K(X)$, and $Y$ and $Z$ any two closed subschemes of $X$, the set of integers $m$ with $\sigma^m(Z ) \subseteq Y$ is as above. The paper presents examples showing that this result may fail to hold if the affine scheme $X$ is not of finite type, or if $X$ is of finite type but the field $K$ has positive characteristic. Keywords:automorphisms, endomorphisms, affine space, commutative algebras, Skolem-Mahler-Lech theoremCategories:11D45, 14R10, 11Y55, 11D88

2. CJM 2007 (vol 59 pp. 186)

Okoh, F.; Zorzitto, F.
 Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields Purely simple Kronecker modules ${\mathcal M}$, built from an algebraically closed field $K$, arise from a triplet $(m,h,\alpha)$ where $m$ is a positive integer, $h\colon\ktil\ar \{\infty,0,1,2,3,\dots\}$ is a height function, and $\alpha$ is a $K$-linear functional on the space $\krx$ of rational functions in one variable $X$. Every pair $(h,\alpha)$ comes with a polynomial $f$ in $K(X)[Y]$ called the regulator. When the module ${\mathcal M}$ admits non-trivial endomorphisms, $f$ must be linear or quadratic in $Y$. In that case ${\mathcal M}$ is purely simple if and only if $f$ is an irreducible quadratic. Then the $K$-algebra $\edm\cm$ embeds in the quadratic function field $\krx[Y]/(f)$. For some height functions $h$ of infinite support $I$, the search for a functional $\alpha$ for which $(h,\alpha)$ has regulator $0$ comes down to having functions $\eta\colon I\ar K$ such that no planar curve intersects the graph of $\eta$ on a cofinite subset. If $K$ has characterictic not $2$, and the triplet $(m,h,\alpha)$ gives a purely-simple Kronecker module ${\mathcal M}$ having non-trivial endomorphisms, then $h$ attains the value $\infty$ at least once on $\ktil$ and $h$ is finite-valued at least twice on $\ktil$. Conversely all these $h$ form part of such triplets. The proof of this result hinges on the fact that a rational function $r$ is a perfect square in $\krx$ if and only if $r$ is a perfect square in the completions of $\krx$ with respect to all of its valuations. Keywords:Purely simple Kronecker module, regulating polynomial, Laurent expansions, endomorphism algebraCategories:16S50, 15A27

3. CJM 1998 (vol 50 pp. 719)

Göbel, Rüdiger; Shelah, Saharon
 Indecomposable almost free modules---the local case Let $R$ be a countable, principal ideal domain which is not a field and $A$ be a countable $R$-algebra which is free as an $R$-module. Then we will construct an $\aleph_1$-free $R$-module $G$ of rank $\aleph_1$ with endomorphism algebra End$_RG = A$. Clearly the result does not hold for fields. Recall that an $R$-module is $\aleph_1$-free if all its countable submodules are free, a condition closely related to Pontryagin's theorem. This result has many consequences, depending on the algebra $A$ in use. For instance, if we choose $A = R$, then clearly $G$ is an indecomposable `almost free' module. The existence of such modules was unknown for rings with only finitely many primes like $R = \hbox{\Bbbvii Z}_{(p)}$, the integers localized at some prime $p$. The result complements a classical realization theorem of Corner's showing that any such algebra is an endomorphism algebra of some torsion-free, reduced $R$-module $G$ of countable rank. Its proof is based on new combinatorial-algebraic techniques related with what we call {\it rigid tree-elements\/} coming from a module generated over a forest of trees. Keywords:indecomposable modules of local rings, $\aleph_1$-free modules of rank $\aleph_1$, realizing rings as endomorphism ringsCategories:20K20, 20K26, 20K30, 13C10
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