We study the rings R whose injective hull E(R-R) is cyclic, extending and simplifying many of the known results on the subject, and obtaining new ones. For example, we prove that if R is any ring such that E(R-R) is cyclic and Dedekind-finite, then R is right self-injective. Moreover, in this case, the ring R turns out to be right co-Hopfian as well as left and right Hopfian. In particular, if E(R-R) is cyclic, then R is right self-injective in the cases where R is commutative, finite-dimensional, semilocal, or strongly pi-regular. We also investigate the Dedekind-finiteness of U-rings and modules, in particular those with cyclic injective hulls.
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