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On isomorphisms of free topological groups, rings and modules generated by topological spaces. (Russian)
Authors: Arnautov Vladimir
1) the free topological groups $F(\Cal X)$ and $F(\Cal Y)$, generated by spaces $\Cal X$ and $\Cal Y$ respectively, are isomorphic;
2) the free topological Abelian groups $A(\Cal X)$ and $A(\Cal Y)$, generated by the spaces $\Cal X$ and $\Cal Y$ respectively, are isomorphic;
3) the free topological rings $R(\Cal X)$ and $R(\Cal Y)$, generated by the spaces $\Cal X$ and $\Cal Y$ respectively, are isomorphic;
4) the free topological modules $M_{\Cal K}(\Cal X)$ and $M_{\Cal K}(\Cal Y)$ over the topological ring, generated by the spaces $\Cal X$ and $\Cal Y$ respectively, are isomorphic;
5) the rings $\Cal K[\Cal X]$ and $\Cal K[\Cal Y]$ of polynomials over the ring $\Cal K$ from sets of variabls $\Cal X$ and $\Cal Y$ respectivly equipped with the maximal topologies which induce on $\Cal K, \Cal X, \Cal Y$ the given topologies are isomorphic.
The implications $1) \Rightarrow 2) \Leftrightarrow 3) \Rightarrow 4)$ are proved. The implication $4) \Rightarrow 5)$ is proved only for commutative rings. Moreover we give examples which show that 4) does not imply 3) and 5) does not imply 4).
Abstract
Consider the following conditions:1) the free topological groups $F(\Cal X)$ and $F(\Cal Y)$, generated by spaces $\Cal X$ and $\Cal Y$ respectively, are isomorphic;
2) the free topological Abelian groups $A(\Cal X)$ and $A(\Cal Y)$, generated by the spaces $\Cal X$ and $\Cal Y$ respectively, are isomorphic;
3) the free topological rings $R(\Cal X)$ and $R(\Cal Y)$, generated by the spaces $\Cal X$ and $\Cal Y$ respectively, are isomorphic;
4) the free topological modules $M_{\Cal K}(\Cal X)$ and $M_{\Cal K}(\Cal Y)$ over the topological ring, generated by the spaces $\Cal X$ and $\Cal Y$ respectively, are isomorphic;
5) the rings $\Cal K[\Cal X]$ and $\Cal K[\Cal Y]$ of polynomials over the ring $\Cal K$ from sets of variabls $\Cal X$ and $\Cal Y$ respectivly equipped with the maximal topologies which induce on $\Cal K, \Cal X, \Cal Y$ the given topologies are isomorphic.
The implications $1) \Rightarrow 2) \Leftrightarrow 3) \Rightarrow 4)$ are proved. The implication $4) \Rightarrow 5)$ is proved only for commutative rings. Moreover we give examples which show that 4) does not imply 3) and 5) does not imply 4).
Institutul de Matematică Academia de Ştiinţe a Moldovei
str. Academiei 5, Chişinău MD-2028, Moldova
str. Academiei 5, Chişinău MD-2028, Moldova
Contents
- An algorithm for forming the Poincare cycle. (Russian)
- A property of the rings of endomorphisms. (Romanian)
- On solution of problems of integer programming. (Russian)
- Rates of queue length convergence for priority queues with orientation. (English)
- On the commutator of two operators in sandwiches. (Russian)
- Idempotents and the radical with respect to a right ideal. (Russian)
- Independence and nilpotency in algebras. (Russian)
- On isomorphisms of free topological groups, rings and modules generated by topological spaces. (Russian)
- Modeling of self-dual Boolean functions commuting with 1 in a 3-valued extension of provability-intuitionistic logic. (Russian)
- $L\sb 2$ estimates for singularly perturbed onedimensional hyperbolic equations of high order. (English)
- On the group of motions of the four-dimensional hyperbolic space of a $120$-hedron. (Russian)
