Abstract. A module M over a ring R is said to satisfy (P ) if every generating set of M contains an independent generating set. The following results are proved;(1) Let τ = (6τ , .τ ) be a hereditary torsion theory such that 6τ = Mod-R. Then every τ -torsionfree R-module satisfies (P ) if and only if S = R/τ (R) is a division ring.(2) Let K be a hereditary pre-torsion class of modules. Then every module in K satisfies (P ) if and only if eitherFor a right R-module M , a subset X of M is said to be a generating set of M if M = Σ x∈X xR; and a minimal generating set of M is any generating set Y of M such that no proper subset of Y can generate M . A generating set X of M is called an independent generating set if Σ x∈X xR = ⊕ x∈X xR. Clearly, every independent generating set of M is a minimal generating set, but the converse is not true in general. For example, the set {2, 3} is a minimal generating set of Z Z but not an independent generating set.It is well-known that every generating set of a right vector space over a division ring contains a minimal generating set (or a basis). This motivated various interests in characterizing the rings R such that every module in a certain class of right R-modules contains a minimal generating set, or every generating set of each module in a certain class of right R-modules contains a minimal generating set (see, for example, In [2, Theorem 2.3], the authors proved that R is a division ring if and only if every R-module has a basis if and only if every irredundant subset of an R-module is independent. This result can be considered in a more general context of a torsion theory. For an R-module M , M is said to satisfy (P ) if every

A element q of a ring R is called quasi-idempotent element if q 2 = uq for some central unit u of R , or equivalently q = ue , where u is a central unit and e is an idempotent of R. In this paper, we define a ring R is almost quasi-clean if each element of R is the sum of a regular element and a quasi-idempotent element. Several properties of almost-quasi clean rings are investigated. We prove that every quasi-continuous and nonsingular ring is almost quasi-clean. Finally, it is determined the conditions under which the idealization of an R-module M is almost quasi clean.

This paper examines the transfer of n­UJ-rings between A and B in an amalgamated duplication of a ring A along some ideal K of a ring B with a ring homomorphism f:A→B (denoted by A⋈^f K).

In this paper our goal to thoroughly determine the rings in which each non-unit element is a product of a nilpotent and a quasi-idempotent.

The aim of the present paper is to characterize associative rings 𝑅 with unity in which 1 + 𝑒𝑅(1 − 𝑒) = 𝑈(𝑅) in terms of some important class of rings in the literature (for example, NR-rings, UU-rings, UJ-rings, UR-rings, exchange rings, 2-primal rings), where 𝑒 2 = 𝑒 ∈ 𝑅 and 𝑈(𝑅) is the set of units of 𝑅.