Abstract: 

It is shown that every commutative sequentially bornologically complete Hausdorff 
algebra $A$ with bounded elements is representable in the form of an (algebraic) inductive limit 
of an inductive system of locally bounded Fr\'echet algebras with continuous monomorphisms if the 
von Neumann bornology of $A$ is pseudoconvex. Several classes of topological algebras $A$ for 
which $r_A(a)\leq \beta_A(a)$ or $r_A(a)= \beta_A(a)$ for each $a\in A$ are described.

1. Institute of Pure Mathematics
   University of Tartu
   2 Liivi St., Room 614
   50409 Tartu, Estonia