Abstract
We investigate situations in which compatible orders on rings with idempotent elements can be extended to more restrictive orders without assuming that the rings are commutative or that they contain multiplicative identities. We show that a ring with a certain kind of irreducible idempotent element is D∗ (i.e., every compatible partial order can be extended to a lattice order that makes the ring a d-ring) if and only if it is O∗ (i.e., every compatible partial order can be extended to a compatible total order); we characterize D∗-algebras over \(\mathbb Q\) that have no identities and are O∗, and we find a class of D∗-algebras that contain subalgebras that are not D∗.
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The authors dedicate this paper to Charles Holland and Jorge Martinez.
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Ma, J., Redfield, R.H. Extending Orders on Rings with Idempotents and d-elements. Order 39, 309–322 (2022). https://doi.org/10.1007/s11083-021-09575-2
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DOI: https://doi.org/10.1007/s11083-021-09575-2