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If H is a maximal cone of a ring A such that the subring generated by H is a commutative integral domain that satisfies a certain centrality condition in A, then there exist a maximal cone H' in a division ring A' and an order preserving monomorphism of A into A', where the subring of A' generated by H' is a subfield over which A' is algebraic. Hypotheses are strengthened so that the main theorems of the author's earlier paper hold for maximal cones. The terminology of the author's earlier paper [3] will be used. For a subsemiring H of a ring A, we write H-H for {x-y:x, y E H}; this is the subring of A generated by H. We say that H is a u-hemiring of A if H is maximal in the class of all subsemirings of A that do not contain a given element u in the center of A. We call H left central if for every a E A and h E H there exists h' E H with ah=h'a. Recall that H is a cone if H`(-H)={O}, and a maximal cone if H is not properly contained in another cone. First note that in [3, Theorem 1] the commutativity of the hemiring, established at the beginning of the proof, was only exploited near the end of the proof and was not used to simplify the earlier details. In the present paper, this fact is put to good use, for commutativity will not be at our immediate disposal. We will show that with stronger hypotheses, the conic hemiring of Theorems 1, 2, and 3 in [3] can be replaced by a maximal cone to obtain Theorems 1, 2, and 3 listed below. This is of interest since, as observed in [2], maximal cones are somewhat more plentiful than conic hemirings. Our first theorem is of a more general nature. THEOREM A. Let H be a nontrivial maximal cone of a ring A such that for every nonzero a, b E H-H, ab is also nonzero. Then thefollowing hold: (1) If H is left central in A, then there exist a conic 1-hemiring H' in a ring A' with 1 and an order preserving monomorphism A--A', where every nonzero element of H' has an invlerse in H' and every nonzero element of A has a left inverse. Received by the editors November 18, 1971. AMS (MOS) subject class/flcations (1970). Primary 06A 10, 06A70; Secondary 16A40.

A note on embedding a partially ordered ring in a division algebra