Injective property relative to nonsingular exact sequences
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
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.
Abstract
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.
참고문헌 (15)
- F.W. Anderson, "Rings and Categories of Modules" (1992), Springer-Verlag
- Sh. Asgari, "t-Extending modules and t-Baer modules", Comm. Algebra, 39(5), pp.1605~1623 (2011)
- Sh. Asgari, "t-Semisimple modules and t-semisimple rings", Comm. Algebra, 41(5), pp.1882~1902 (2013)
- N. V. Dung, "Extending Modules" (1994), Longman
- C. Faith, "When are proper cyclics injective?", Pacific J. Math, 45, pp.97~112 (1973)
- K. R. Goodearl, "Singular Torsion and Splitting Properties" (1972), AMS
- K. R. Goodearl, "An Introduction to Noncommutative Noether-ian Rings" (2004), Cambridge University Press
- S. K. Jain, "Cyclic Modules and the Structure of Rings" (2012), Oxford University Press
- T. Y. Lam, "Lectures on Modules and Rings" (1998), Springer-Verlag
- G. Lee, "Modules whose endomorphism rings are von Neumann regular", Comm. Algebra, 41(11), pp.4066~4088 (2013)
- W. K. Nicholson, "Quasi-Frobenius Rings" (2003), Cambridge University Press
- B. L. Osofsky, "Homological properties of rings and modules", Doctor, pp.1964
- B. L. Osofsky, "Cyclic modules whose quotients have all complements submodules direct summands", J. Algebra, 139(2), pp.342~354 (1991)
- S. T. Rizvi, "On continuous and singular modules", Noncommutative Ring Theory, Proc., Athens, Lecture Notes in Mathematics, 1448, pp.116~124 (1990), Springer Verlag
- M. F. Yousif, "On pseudo-Frobenius rings", Canad. Math. Bull, 48(2), pp.317~320 (2005)