Ordered Rings- Part III

Summary. This series of papers is devoted to the notion of the ordered ring, and one of its most important cases: the notion of ordered field. It follows the results of [6]. The idea of the notion of order in the ring is based on that of positive cone i.e. the set of positive elements. Positive cone has to contain at least squares of all elements, and has to be closed under sum and product. Therefore the key notions of this theory are that of square, sum of squares, product of squares, etc. and finally elements generated from squares by means of sums and products. Part III contains the classification of products of such elements. MML Identifier: O RING 3. The papers [1], [2], [7], [3], [4], and [5] provide the terminology and notation for this paper. In the sequel R will denote a field structure and x, y will denote scalars of R. Next we state a number of propositions: (1) If x is a square and y is a square, then x · y is a product of squares. (2) If x is a product of squares and y is a square, then x · y is a product of squares. (3) If x is a square and y is a product of squares or x is a square and y is a amalgam of squares, then x · y is a amalgam of squares. (4) If x is a product of squares and y is a product of squares or x is a product of squares and y is a amalgam of squares, then x ·y is a amalgam of squares. (5) If x is a amalgam of squares and y is a square or x is a amalgam of squares and y is a product of squares or x is a amalgam of squares and y is a amalgam of squares, then x · y is a amalgam of squares. (6) If x is a square and y is a sum of squares or x is a square and y is a sum of products of squares or x is a square and y is a sum of amalgams of squares or x is a square and y is generated from squares, then x · y is generated from squares