Then I contains an

invertible element of R, and so I = R = [M.

14) and a topological invertible element is said to be proper (see [34], p.

r](A)) the set of all topologically left (right) invertible elements in A and by [G.

The proof for right topological invertible elements is similar.

We say that a left (right) ideal I in A is a topological left (respectively, right) ideal if I does not contain left (respectively, right) topologically invertible elements.

Every graded connected bialgebra B is in fact a Hopf algebra [5]--this means, by definition, that the identity map id : B [right arrow] B is an

invertible element in the convolution algebra L(B, B).

Remark: There are many other families of the Renner monoids R with group W of

invertible elements in R ([5,7,8]).

Later Araujo ([1]) made a study of these maps in his thesis and has characterized Banach-Stone maps T , as those isometric isomorphisms which take

invertible elements to

invertible elements or maximal ideals to maximal ideals etc.

Traditionally, RG will denote the group ring of G over R with group S(RG) consisting of all normalized

invertible elements of orders which are powers of p.

Let TqinvA denote the set of all topologically quasi-

invertible elements in A, QinvA the set of all quasi-

invertible elements in A and, for a unital topological algebra A, let TinvA denote the set of all topologically

invertible elements in A and InvA the set of all

invertible elements in A.

Later Araujo ([1]) made a study of these maps in his thesis and has characterized Banach-Stone maps T, as those isometric isomorphisms which take

invertible elements to

invertible elements or maximal ideals to maximal ideals etc.

This assumption and the first equivalence of (4) imply that we have to study the following three cases: 1) p and s are even ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), which implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; 2) p and s are odd ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the group of

invertible elements of the ring [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), one of the numbers q and r is odd; 3) p and s are odd, q and r are even ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).