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Dergiler / Turkish Journal of Mathematics / 2014 / Cilt: 38 - Sayı: 4

Some rings for which the cosingular submodule of every module is a direct summand

Derya TÜTÜNCÜ KESKIN
Rachid TRIBAK
Nil ERTAS ORHAN
Patrick SMITH F.
Sayfa
649–657
Yayın Tarihi
2022-11-05
DOI
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

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.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:The submodule \overline{Z}(M) = \cap {N | M/N is small in its injective hull} was introduced by Talebi and Vanaja in 2002.
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Temel Bulgular & Veri:A ring R is said to have property (P) if \overline{Z}(M) is a direct summand of M for every R-module M.
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Sonuç & Katkı: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.
🔬Tür:Araştırma Makalesi
📍Kapsam:2002 Verileri
⚙️Değişkenler:von Neumann regular ring · perfect ring · (non)cosingular submodule

Özet

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.

Abstract

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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Bu Makaleye Atıf Yap

KESKIN, D. T., TRIBAK, R., ORHAN, N. E., F., P. S. (2022). Some rings for which the cosingular submodule of every module is a direct summand. Turkish Journal of Mathematics, Cilt: 38 - Sayı: 4, 649–657. https://ajindex.com/articles/608243

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Kaynakça (10)

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