Exercise 12, § 6

A subset $M$ of a topological ring is said to be right bounded (resp. left bounded) if, for each neighbourhood $U$ of $o$ in $A$, there is a neighbourhood $V$ of $o$ such that $VM \subset U$ (resp. $MV \subset U$); $M$ is bounded if it is both left and right bounded. The topology of $A$ is said to be locally bounded (and $A$ is said to be a locally bounded topological ring) if there is a bounded neighbourhood of $o$ in $A$.

a) Every topological ring which has a fundamental system of neighbourhoods of $o$ consisting of right ideals is right bounded. Show that if $I$ is infinite and $B$ is a division ring in Exercise 10 a), then the ring $A$ is left bounded, but there is no right bounded neighbourhood of $o$ in $A$.

b) If a topological ring $A$ is right bounded (resp. bounded) and has a fundamental system of neighbourhoods of $o$ which are subgroups of the additive group of $A$, then it has a fundamental system of neighbourhoods of $o$ which are right (resp. two-sided) ideals.

c) Every finite union of bounded sets is bounded; the closure of a bounded set is bounded; if $M$ and $N$ are bounded, then so are $M + N$ and $MN$.

d) Every precompact set in a topological ring $A$ is bounded.

e) If $M$ and $N$ are two bounded subsets of $A$, show that the mapping $(x, y) \to xy$ elements of $M \times N$ into $A$ is uniformly continuous.

f) Let $A$ be a Hausdorff topological ring which has an identity element. Show that if $(x_n)$ is a sequence of units which tends to $o$ in $A$, then the set of elements $x_n^{-1}$ is neither left nor right bounded.

g) If $A$ is a bounded ring with an identity element, show that the mapping $x \to x^{-1}$ is uniformly continuous on the set $A^*$ of units of $A$.

h) Show that if $A$ is a complete Hausdorff bounded ring with an identity element, then the set $A^*$ of units and the radical of $A$ are closed [use g)].

i) A subset of a product $\prod_i A_i$ of topological rings is bounded if and only if all its projections are bounded.

j) Every subring of a locally bounded ring and every product of locally bounded rings are locally bounded rings.

k) The completion of a locally bounded Hausdorff ring is locally bounded.