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We investigate modules $M$ having the injective property relative to nonsingular modules. Such modules are called ``$\mathcal N$-injective modules''. It is shown that $M$ is an $\mathcal N$-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 $\mathcal N$-injective $R$-module is injective precisely when $R$ is a right nonsingular ring. We prove that the endomorphism ring of an $\mathcal N$-injective module has a von Neumann regular factor ring. Every (finitely generated, cyclic, free) $R$-module is $\mathcal N$-injective, if and only if $R^{(\mathbb N)}$ is $\mathcal N$-injective, if and only if $R$ is right $t$-semisimple. The $\mathcal N$-injective property is characterized for right extending rings, semilocal rings and rings of finite reduced rank. Using the $\mathcal N$-injective property, we determine the rings whose all nonsingular cyclic modules are injective.
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참고문헌 (15)
참고문헌 신청
B. L. Osofsky / 1964 / Homological properties of rings and modules / Doctor / Rutgers University
B. L. Osofsky / 1991 / Cyclic modules whose quotients have all complements submodules direct summands / J. Algebra 139 (2) : 342 ~ 354
C. Faith / 1973 / When are proper cyclics injective? / Pacific J. Math 45 : 97 ~ 112
F.W. Anderson / 1992 / Rings and Categories of Modules / Springer-Verlag
G. Lee / 2013 / Modules whose endomorphism rings are von Neumann regular / Comm. Algebra 41 (11) : 4066 ~ 4088
B. L. Osofsky / 1991 / Cyclic modules whose quotients have all complements submodules direct summands / J. Algebra 139 (2) : 342 ~ 354
C. Faith / 1973 / When are proper cyclics injective? / Pacific J. Math 45 : 97 ~ 112
F.W. Anderson / 1992 / Rings and Categories of Modules / Springer-Verlag
G. Lee / 2013 / Modules whose endomorphism rings are von Neumann regular / Comm. Algebra 41 (11) : 4066 ~ 4088
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