Abstract
The rings in the title were called UN rings by Călugăreanu in [G. Călugăreanu, UN-rings, J. Algebra Appl. 15(9) (2016) 1650182]. He gave two examples of simple UN rings: matrix rings over a skew field and a ring, which is the filtered union of such rings. We give new examples of simple UN rings as endomorphism rings of ‘vector space like’ modules and determine the structure of UN rings, which satisfy a polynomial identity or have Krull dimension. We also answer some questions in [G. Călugăreanu, UN-rings, J. Algebra Appl. 15(9) (2016) 1650182] about Morita equivalence of UN rings and show that this question is related to Köthe’s conjecture. Finally a complete characterization is given of modules over a Dedekind domain (in particular Abelian groups) and modules of finite length with a UN endomorphism ring.
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