Some rings for which the cosingular submodule of every module is a direct summand
The submodule \overline{Z}(M) = \cap {N | M/N is small in its injective hull} was introduced by Talebi and Vanaja in 2002. A ring R is said to have property (P) if \overline{Z}(M) is a direct summand of M for every R-module M. It is shown that a commutative perfect ring R has (P) if and only if R is semisimple. An example is given to show that this characterization is not true for noncommutative rings. We prove that if R is a commutative ring such that the class {M \in Mod-R | \overline{Z}R(M) = 0} is closed under factor modules, then R has (P) if and only if the ring R is von Neumann regular.
Anahtar Kelimeler:
von Neumann regular ring, perfect ring, (non)cosingular submodule
Some rings for which the cosingular submodule of every module is a direct summand
The submodule \overline{Z}(M) = \cap {N | M/N is small in its injective hull} was introduced by Talebi and Vanaja in 2002. A ring R is said to have property (P) if \overline{Z}(M) is a direct summand of M for every R-module M. It is shown that a commutative perfect ring R has (P) if and only if R is semisimple. An example is given to show that this characterization is not true for noncommutative rings. We prove that if R is a commutative ring such that the class {M \in Mod-R | \overline{Z}R(M) = 0} is closed under factor modules, then R has (P) if and only if the ring R is von Neumann regular.
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- ISSN: 1300-0098
- Yayın Aralığı: Yılda 6 Sayı
- Yayıncı: TÜBİTAK
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