The extension R⊆S is said to have FIP if [R, S] is finite. The extension R⊆S is said to have FCP if each chain of R-sub algebras of S is finite. Several parallel characterizations of the FIP and FCP properties are given. Also a number of results about FCP are generalized from domain to arbitrary (commutative) rings. Let R⊆S be rings, with ̅R the integral closure of R in S. Then R⊆S satisfies FIP (resp. , FCP) if and only if both R ⊆ ̅R and ̅R ⊆S satisfy FIP (resp. ,FCP) If R integrally closed in S, then R⊆S satisfies FIP if and only if R⊆S satisfies FCP if and only if (R, S) is a normal pair such that Supp(S/ R) is finite. If R⊆S is integral and has conductor C, then R⊆S satisfies FCP if and only if S is a finitely generated R- module such that R / C is an Artinian ring. The characterizations of FIP and FCP for integral extensions feature natural roles for the intermediate rings arising from seminormalization and t-closure.

In commutative Algebra we know that a commutative ring R is Noetherian iff every prime ideal of R is finitely generated (Cohen’s Theorem). Also a commutative Noetherian ring R is a principal ideal ring iff every maximal ideal of R is principal (Kaplansky’s Theorem). So with composition these theoremes we have the following theorem in commutative rings that a commutative ring R is a principal ideal ring iff every prime ideal of R is principal (Kaplansky – Cohen Theorem). The Goal of this thesis is a generalization of this theorem and theorems of Cohen and Kaplansky for noncommutative rings. In this case we are defineing competely prime ideals and, then with replace competely prime ideals in a noncommutative ring with prime ideals in a commutative ring, we show a generalization of these theorems in noncommutative rings. Also we are defineing a point annihilator set and, with this set, we are a generalization of theorem of Cohen - Kaplansky in noncommutative rings. Finally we show that if R be a semiprime ring with r. K. dim(R) ≤ 1, then R is a principal right ideal ring iff its maximal right ideals are principal.

 

 For a pair of commutative rings With the same identity, S is a minimal ring extension of R if there are no rings properly between R and S. Such an extension is said to be closed if R is integrally closed in S; otherwise, S is integral over R and the extension is a minimal integral extension. For a minimal integral extension , the conductor is a maximal ideal of R. If M has no nonzero annihilators in S, then S is isomorphic to an overring of R in . The complete ring of quotients of R. conversely, if M has a nonzero annihilator in S, then S cannot be isomorphic to an overring of R . A complete characterization of minimal overrings is given in the case R is a reduced ring. Also examples are given to illustrate the various types of minimal integral extensions.

Given a family of commutative rings, we investigate the family
of Artinian subrings of
This thesis deals also with the question
of when a pair
is zero-dimensional, where is a family of zerodimensional
pairs.

Abstract:
In this thesis, first the unique factorization domains and atomic domains are in-
vestigated. A factorization domain R is called half-factorial domain (HFD), if for each non-zero non-unit element a₁a₂…a_n = b₁b₂…b_m in which a_i and b_j are irreducible in R, then n=m. Also factorization domain R is called OHFD, if a₁a₂…a_n= b₁b₂…b_n in which a_i and b_i are irreducible in R, then each element of a_i are associate with at least one of b_j elements. We show that any OHFD is a UFD.
 

Let R be a reduced commutative neotherian ring. We provide conditions equivalent to isomorphism for completely decomposable finitely generated modules over R. We show that, if R is one dimensional and R satisfies the Krull-Schmidt property for ideals, then any overring of R must also have this property. We also show that if R is both local and one dimensional, satisfying the Krull-Schmidt property for direct sums of rank one modules

In this thesis all rings are unitary commutative rings. If R is a ring and b in the R is not unite, then b is called a pseudo- irreducible element if it is impossible to factor b as b=cd with c and d comaximal non- units. A non-unit element b has a complete comaximal factorization if we can write b=b1b2........bm such that bi are pairwise comaximal factorization domain (CFD) if every nonzero non-unit element of R has a complete comaximal factorization . We show that every noetherian domain is CFD. We find conditions for where CFD are UCFD, and also condition to share the CFD property frome R to R[x]. Also we obtain conditions for where CFD property can be shared from R to R/P for a prime ideal P of R. Finally we generalize the factorization to ideals instead of elements

The notion of Rickart modules was defined recently. It has been shown that a direct sum of Rickart modules is not a Rickart module, in general. In this paper we investigate the question: When are the direct sums of Rickart modules, also Rickart? We show that ifM_iis M_j –injective for alli<j ∈I={1 ,2 ,…,n } then ⨁_(i=1)^n M_iis a Rickart module if and only if M_i is M_j-Rickartfor all i<j ∈I . As a consequence we obtain that for a nonsingular extending module M, E(M)⨁Mis always a Rickart module. Other characterizations for direct sums to be Rickart under certain assumptions are provided. We also investigate when certain classes of free modules over a ring R, are Rickart. It is shown that every finitely generated free R-module is Rickart precisely when R is a right semihereditary ring. As an application, we show that a commutative domain R is Prufer if and only if the free R-module R(2) is Rickart. We exhibit an example
of a module M for which M(2) is Rickart but M(3) is not so. Further, von Neumann regular rings are characterized in terms of Rickart modules. It is shown that the class of rings R for which every finitely cogenerated right R-module is Rickart, is precisely that of right V -rings.
 

Let R be any ring ; is called a weak zero-divior if there are with . It is shown that , in any ring R , the element of minimal prime ideal are weak zero-divior .Examples show that a minimal prime ideals may have elements which are neither left nor right zero - divisors one of right zero-divisors.However, every R has a minimal prime ideal consisting of left zero-divisors and one of right zero-divisors . The union of the minimal prime ideals is studied in 2-primal rings and the union of the minimal strongly prime ideals (in the sense of rowen ) in NI-rings.