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

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