ON RINGS IN WHICH EVERY IDEAL IS WEAKLY PRIME

Rings for which every simple module is almost injective

We introduce the class of “right almost V-rings” which is properly between the classes of right V-rings and right good rings. A ring R is called a right almost V-ring if every simple R-module is almost injective. It is proved that R is a right almost V-ring if and only if for every R-module M, any complement of every simple submodule of M is a direct summand. Moreover, R is a right almost V-rin...

Commutative rings in which every finitely generated ideal is quasi-projective

This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. Section 2 provides some preliminaries on quasi-projective modules over commutative rings. Section 3 investigates the correlation with well-known Prüfer conditions; namely, we prove that this class of rings stands strictly between the two classes of arithmetical...

Rings of Continuous Functions in Which Every Finitely Generated Ideal is Principal

rings for which every simple module is almost injective

we introduce the class of “right almost v-rings” which is properly between the classes of right v-rings and right good rings. a ring r is called a right almost v-ring if every simple r-module is almost injective. it is proved that r is a right almost v-ring if and only if for every r-module m, any complement of every simple submodule of m is a direct summand. moreover, r is a right almost v-rin...

When every $P$-flat ideal is flat

In this paper‎, ‎we study the class of rings in which every $P$-flat‎ ‎ideal is flat and which will be called $PFF$-rings‎. ‎In particular‎, ‎Von Neumann regular rings‎, ‎hereditary rings‎, ‎semi-hereditary ring‎, ‎PID and arithmetical rings are examples of $PFF$-rings‎. ‎In the context domain‎, ‎this notion coincide with‎ ‎Pr"{u}fer domain‎. ‎We provide necessary and sufficient conditions for‎...