PREORDERINGS ON LOCAL GLOBAL RINGS

Suppose A is a local global ring (with many units) and $T{\subset}A$ is a preordering. Let $a_i{\in}A^*$, $i=1,2,{\cdots},n$ and $a{\in}({\sum}_{i=1}^{l-1}\;a_iT){\cap}A^*$. Then, for any integer l, 1 < l ${\leq}$ n, there exist $x{\in}({\sum}_{i=1}^{l-1}\;a_iT){\cap}A^*$ and $y{\in}({\sum}_{i=l}^n\;a_iT){\cap}A^*$ such that a=x+y.