In a fully ordered (f.o.) ring with identity, the set of all bounded elements as defined below might be an Archimedian subring. Most of the examples of f.o. rings constructed in literature having the bounded set as Archimedian subring are polynomial rings. For example I[x], R[x] etc., where I is the ring of integers and R is the field of rationals, with lexicographic ordering. Now we ask whether a f.o. ring with identity, with the set of bounded elements as Archimedian subring can be a polynomial ring over an Archimedian subring. This is answered affirmatively in Theorem 1. It is proved in Theorem 3 that f.o. rings with identity and with every positive element a large element, belong to the above class. The problem then arises as to when the set of all bounded elements, called a weak Archimedian sub- ring in [2], becomes an Archimedian subring. This problem is completely solved in Theorem 2. The concept of weak Archimedian rings is found to be useful by the author in characterizing some f.o. rings as algebraic algebras in [3].