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Keywords:
reflexive ring; $\mathcal Z$-reflexive ring; $J$-reflexive ring; central reflexive ring; singular ideal
reflexive ring; $\mathcal Z$-reflexive ring; $J$-reflexive ring; central reflexive ring; singular ideal
Summary:
We introduce the notion of $\mathcal Z$-reflexive rings to describe reflexivity of rings in terms of their singular ideals. We show that $\mathcal Z$-reflexive ring is proper common generalization of a central reflexive ring, $\mathcal Z$-reversible ring, and singular clean ring. We discuss some its properties, characterizations, and relations with some extension rings. We show that a ring $R$ is right $\mathcal Z$-reflexive if and only if $M_n(R)$ is right $\mathcal Z$-reflexive for every positive integer $n$. Also, we share the connection of right $\mathcal Z$-reflexive rings with $J$-reflexive rings.
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