c-Orderable division rings with involution

M Chacron
1982 Journal of Algebra  
DEDICATEDTO THE MEMORY OF MY MOTHER. Let R be any division ring with involution. The *-core (resp. core) of R is the set of elements of the form where each term pi is some non-zero product of norms aa* (resp. squares a'). T. Szele proved that in order for the division ring R to be Hilbert ordered (e.g., R has some total order relation, which is additive and multiplicative) it is necessary and sufficient that the core of R exclude 0. In this paper we shall investigate the division rings with
more » ... lution R such that the *-core of R excludes 0. We call these division rings c-orderable. In fact, any c-orderable division ring R is shown to admit an ordering of the following type. For a, b, c, d arbitrary elements of R: we have (1) a>b implies a*>b*, (2) a > b and c > d imply a + c > b + d, (3) abwhenevera=a"andb=b*,and (4) a > b implies uxx* >, bxx*. Such an ordering we call a c-ordering (c for *-core). Among other things, we show that this ordering is conditionally Baer, that is, (5) s = s* > 0 implies XXX* > 0, provided s can be bounded below and above by positive rationals. Whether or not this proviso can be dropped is an open question. Another important property of the c-ordering is that the set of bounded elements x at this ordering (e.g., XX* < rO for some rational r") is a *-valuation subring Y
doi:10.1016/0021-8693(82)90053-9 fatcat:5zmz3pjtcbcg3jjfjdbit4hopi