We investigate modules M having the injective property relative to nonsingular modules. Such modules are called "N -injective modules". It is shown that M is an N -injective R-module if and only if the annihilator of Z 2 (R R ) in M is equal to the annihilator of Z 2 (R R ) in E(M ). Every N -injective R-module is injective precisely when R is a right nonsingular ring. We prove that the endomorphism ring of an Ninjective module has a von Neumann regular factor ring. Every (finitely generated,

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... clic, free) R-module is N -injective, if and only if R (N) is N -injective, if and only if R is right t-semisimple. The N -injective property is characterized for right extending rings, semilocal rings and rings of finite reduced rank. Using the N -injective property, we determine the rings whose all nonsingular cyclic modules are injective. 559 560 M. ARABI-KAKAVAND, SH. ASGARI, AND Y. TOLOOEI Clearly, a submodule A of M is Z 2 -torsion if and only if A ≤ Z 2 (M ). The class of Z 2 -torsion modules is closed under submodules, factor modules, direct sums, and extensions. In In fact, t-extending modules are precisely the modules M for which every closed submodule of M containing Z 2 (M ) is a direct summand of M . Over the last 50 years numerous mathematicians have investigated rings over which certain cyclic modules have a homological property. Among these, determining the rings whose certain cyclic modules are injective has been of interest. Osofsky [12] proved that every cyclic R-module is injective, if and only if every R-module is injective, if and only if R is semisimple. A cyclic R-module is called proper cyclic if it is not isomorphic to R. A ring R is called a right PCI-ring if every proper cyclic R-module is injective. Faith [5] proved that a right PCI-ring is either a semisimple ring or a simple right semihereditary right Ore domain. An excellent reference for a thorough study of these rings is [8] . The rings for which every singular module is injective were studied by Goodearl [6]. He called them right SI-rings and characterized such rings as those nonsingular ones for which R/I is semisimple for every essential right ideal I of R. Osofsky and Smith [13] showed that every singular cyclic Rmodule is injective if and only if R is a right SI-ring. More results on such rings can be found in [4] and [14]. Motivated by these, a natural question is: "What are the rings whose all nonsingular cyclic modules are injective?" In [3] the rings whose all nonsingular modules are injective were studied. Such rings are called right t-semisimple rings. It was shown that R is right t-semisimple, if and only if every nonsingular R-module is semisimple, if and only if R/Z 2 (R R ) is a semisimple ring, if and only if R is a direct product of two rings, one is semisimple and the other is right Z 2 -torsion. By [3, Example 4.15], the class of right t-semisimple rings is properly contained in that of rings R for which every nonsingular cyclic R-module is injective. This raises another question: "Under which condition(s) the class of rings R for which every nonsingular cyclic R-module is injective coincides with that of right t-semisimple rings?" But, it is a fact, obtained by Baer's criterion, that a nonsingular R-module M is injective precisely when M is injective relative to the nonsingular R-module R/Z 2 (R R ). This leads us to investigate the modules M which are injective relative to nonsingular modules for finding the answers of the above questions. Let M and L be R-modules. Recall that M is said to be L-injective (or, injective relative to L) if for every monomorphism f : K → L and every homomorphism g : K → M , there is a homomorphism h : L → M such that hf = g. We say that an R-module M is N -injective if M is injective relative