Commutative semi-coherent and semi-regular rings
Let R be a commutative ring with identity. R is said to be coherent if all finitely generated ideals of R are finitely presented. R is said to be reduced if it has no non-zero nilpotent elements. Let min R denote the spectrum of minimal prime ideals of R. An ideal \(I\neq 0\) of R is said to be a non-essential ideal if there exists an ideal \(J\neq 0\) of R such that \(I\cap J=0.\) The author gives the definitions: \((1)\quad R\quad is\) said to be semi- coherent if \(Hom_ R(B,C)\) is a submodule of a flat R-module whenever B and C are injective R-modules; \((2)\quad R\quad is\) said to be semi- regular if every R-module is a submodule of a flat module; \((3)\quad An\) R-module A is said to be semi-injective if every R-homomorphism from a finitely generated ideal of R to A can be lifted to an R-homomorphism from R to A. Some of the results given by the author are: \((1)\quad A\) reduced ring R is semi-coherent if and only if min R is compact; \((2)\quad If\) R is semi-coherent, then a finitely generated submodule of a free R-module is flat if and only if it is projective; \((3)\quad If\) the total ring of quotients of R is semi-regular, then R is a semi-coherent ring; \((4)\quad R\quad is\) semi-regular if and only if R is coherent and \(R_ M\) is semi-regular for every maximal ideal M of R; \((5)\quad If\) R is Noetherian, then R is semi-regular if and only if R is self-injective; \((6)\quad Let\) R be an integral domain. Then R is a Prüfer domain if and only if R/I is a semi-regular ring for every non-zero, finitely generated ideal I of R; \((7)\quad The\) following statements are equivalent: \((a)\quad R\quad is\) semi-regular. \((b)\quad An\) R-module is flat if and only if it is semi-injective. \((c)\quad R\quad is\) coherent and a semi-injective R-module. \((d)\quad R\quad is\) coherent and has the double annihilator condition on finitely generated ideals. \((e)\quad R\quad is\) coherent and every finitely presented R-module is reflexive; \((8)\quad If\) R is semi-regular, then the non-essential prime ideals of R are both maximal and minimal prime ideals and are direct summands of R.
scientific article; zbMATH DE number 3959602
scientific article