By studying and using the quasi-pure part concept, we im-prove some statements and show that some assumptions in some articles are super uous. We give some characterizations of Gelfand RINGS. For example: we prove that R is Gelfand if and only if m (Σ,, 2A I,) Σ,= , 2A m(I, ), for each family fI, g, 2A of ideals of R, in addition if R is semiprimitive and Max(R) ,Y ,Spec(R), we show that R is a Gelfand ring if and only if Y is normal. We prove that if R is reduced ring, then R is a von Neumann regular ring if and only if Spec(R) is regular. It has been shown that if R is a Gelfand ring, then Max(R) is a quotient of Spec(R), and sometimes hM(a)'s behave like the zerosets of the space of maximal ideal. Finally, it has been proven that Z ( Max(C(X)) ) = fhM(f): f 2 C(X)g if and only if fhM(f): f 2 C(X)g is closed under countable intersection if and only if X is pseudocompact.

It is shown that a COMMUTATIVE reduced ring R is a Baer ring if and only if it is a CS-ring; if and only if every dense subset of Spec (R) containing Max (R) is an extremally disconnected space; if and only if every non-zero ideal of R is essential in a principal ideal generated by an idempotent.

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Let $R$ be a COMMUTATIVE ring with non-zero identity, and $\delta :\mathcal{I(R)}\rightarrow\mathcal{I(R)}$ be an ideal expansion where $\mathcal{I(R)}$ is the set of all ideals of $R$. In this paper, we introduce the concept of $\delta$-$n$-ideals which is an extension of $n$-ideals in COMMUTATIVE RINGS. We call a proper ideal $I$ of $R$ a $\delta$-$n$-ideal ifwhenever $a,b\in R$ with$\ ab\in I$ and $a\notin\sqrt{0}$, then $b\in \delta(I)$. For example, an ideal expansion $\delta_{1}$ is defined by $\delta_{1}(I)=\sqrt{I}.$ In this case, a $\delta_{1}$-$n$-ideal $I$ is said to be a quasi $n$-ideal or equivalently, $I$ is quasi $n$-ideal if $\sqrt{I}$ is an $n$-ideal. A number of characterizations and results with manysupporting examples concerning this new class of ideals are given. In particular, we present some results regarding quasi $n$-ideals. Furthermore, we use $\delta$-$n$-ideals to characterize fields and UN-RINGS.

In this paper, we introduce a new class of modules that is closely related to the class of Noetherian modules. Let R be a COMMUTATIVE ring with identity, and M be an R-module such that Nil(M) is a divided prime submodule of M. M is called a nonnil-Noetherian R-module if every nonnil submodule of M is  nitely-generated. We prove that many properties of the Noether-ian modules are also true for the nonnil-Noetherian modules.

In this paper, we introduce the concept of uniformly $n$-ideal ofCOMMUTATIVE RINGS which is a special type of $n$-ideal. We call aproper ideal $I$ of $R$ a uniformly $n$-ideal if there exists apositive integer $k$ for $a,b\in R$ whenever $ab\in I$ and$a\notin I$ implies that $b^{k}=0.$ The basic properties ofuniformly $n$-ideals are investigated in detail. Moreover, somecharacterizations of uniformly $n$-ideals are obtained for somespecial RINGS.

The current work extends the class of COMMUTATIVE MTL-RINGS established by the authors to the non-COMMUTATIVE ones. That class of RINGS will be named generalized MTL-RINGS since they are not necessary COMMUTATIVE. We show that in the non-COMMUTATIVE case, a local ring with identity is a generalized MTL-ring if and only if it is an ideal chain ring. We prove that the ring of matrices over an MTL-ring is a non-COMMUTATIVE MTL-ring. We also study their representation in terms of subdirect irreducibility.

The modules (RINGS) satisfying acc on certain submodules investigated in [2] and various important properties of Noetherian modules and RINGS can be generalized to modules and RINGS of this class. The present author introduced and developed the concept of modules (RINGS) satisfying dcc on certain submodules in [5] and [6]. In this paper we present a new characterization of RINGS satisfying dcc on certain submodules.