Abstract
We propose the study of rings, every partial ordering of which extends to a lattice ordering (i*-rings). We show techniques enabling us, in some important cases, to decide whether a ring is or is not an L*-ring. In particular, we show that direct products of rings without nilpotent elements , rings of matrices and polynomial rings are not L*-rings. On the other hand, we give an example of a (non-associative) ring every directed partial order of which extends to a lattice order but fails to extend to a total order (not O*-ring). We also prove that under the requirement that the orders in question are consistent (Definition 2.3) or that squares of elements are positive, an L*-ring can have only algebraic elements. Finally, we generalize the main result of [3] to L*-rings having positive squares.
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