Chains of Rings with Local Formal Fibers

Let (T, M) be a complete local domain containing the integers. Let p ₁ ? p ₂ ? ··· ? p _n be a chain of nonmaximal prime ideals of T such that T _{p n} is a regular local ring. We construct a chain of excellent local domains A _n  ? A _n?1 ? ··· ? A ₁ such that for each 1 ≤ i ≤ n, the completion of A _i is T, the generic formal fiber of A _i is local with maximal ideal p _i , and if I is a nonzero ideal of A _i then A _i /I is complete. We then show that if Q is a nonmaximal prime ideal of T and 1 ≤ h = ht _T Q, then there is a chain of excellent local domains B ₀ ? B ₁ ? ··· ? B _h  ? T such that for every i = 0, 1, 2,…, h we have ht(Q ∩ B _i ) = i, the completion of B _i is isomorphic to T[[X ₁, X ₂,…, X _i ]] where the X _j 's are indeterminants, and the formal fiber of Q ∩ B _i is local.     

Rank One Discrete Valuations of Power Series Fields

In this article we study rank one discrete valuations of the field k((X ₁,…, X _n )) whose center in k[[X ₁,…, X _n ]] is the maximal ideal. In Sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote Section 2 to finding an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in Section 5.

The constructions given in these sections are not effective in the general case, because we need either to use Zorn's lemma or to know explicitly a section σ of the natural homomorphism R _v  → Δ _v between the ring and the residue field of the valuation v.

However, as a consequence of this construction, in Section 7, we prove that k((X ₁,…, X _n )) can be embedded into a field L((Y ₁,…, Y _n )), where L is an algebraic extension of k and the “extended valuation” is as close as possible to the usual order function.     

Completions of UFDs with Semi-Local Formal Fibers

Let (T, M) be a complete local ring such that |T/M| = | T |. Given a finite set of incomparable nonmaximal prime ideals C of T, we provide necessary and sufficient conditions for T to be the completion of a local UFD A, whose generic formal fiber is semilocal with maximal ideals the elements of C. We also show that, given the T above, we can find necessary and sufficient conditions for T to be the completion of a UFD, whose formal fiber over a height one prime ideal is semilocal.

Communicated by I. Swanson.     

On Jordan Left Derivations of Lie Ideals in Prime Rings

Let R be a prime ring with characteristic different from two and U be a Lie ideal of R such that u² U for all u U. In the present paper it is shown that if d is an additive mappings of R into itself satisfying d(u²) = 2ud(u), for all u U, then either U Z(R) or d(U) = (0).1991 Mathematics Subject Classification 16W25 16N60     

On the Attached Prime Ideals of Local Cohomology Modules

In this article, we shall prove some new properties about attached prime ideals over local cohomology modules. Also we generalize some of the results of [2 Brodmann , M. P. , Sharp , R. Y. ( 1998 ). Local Cohomology; An Algebraic Introduction with Geometric Applications . Cambridge : Cambridge University Press .[Crossref] , [Google Scholar]].     

On Generalizations of Prime Ideals

Let R be a commutative ring with identity. Let φ: S(R) → S(R) ∪ {?} be a function, where S(R) is the set of ideals of R. Suppose n ≥ 2 is a positive integer. A nonzero proper ideal I of R is called (n ? 1, n) ? φ-prime if, whenever a ₁, a ₂, ?, a _n  ∈ R and a ₁ a ₂?a _n  ∈ I?φ(I), the product of (n ? 1) of the a _i 's is in I. In this article, we study (n ? 1, n) ? φ-prime ideals (n ≥ 2). A number of results concerning (n ? 1, n) ? φ-prime ideals and examples of (n ? 1, n) ? φ-prime ideals are also given. Finally, rings with the property that for some φ, every proper ideal is (n ? 1, n) ? φ-prime, are characterized.     

Efficient Generation of Prime Ideals in Polynomial Rings up to Radical

We call an ideal I of a ring R radically perfect if among all ideals whose radical is equal to the radical of I, the one with the least number of generators has this number of generators equal to the height of I. Let R be a ring and R[X] be the polynomial ring over R. We prove that if R is a strong S-domain of finite Krull dimension and if each nonzero element of R is contained in finitely many maximal ideals of R, then each maximal ideal of R[X] of maximal height is the J _max-radical of an ideal generated by two elements. We also show that if R is a Prüfer domain of finite Krull dimension with coprimely packed set of maximal ideals, then for each maximal ideal M of R, the prime ideal MR[X] of R[X] is radically perfect if and only if R is of dimension one and each maximal ideal of R is the radical of a principal ideal. We then prove that the above conditions on the Prüfer domain R also imply that a power of each finitely generated maximal ideal of R is principal. This result naturally raises the question whether the same conditions on R imply that the Picard group of R is torsion, and we prove this to be so when either R is an almost Dedekind domain or a Prüfer domain with an extra condition imposed on it.     

Divisibility Theory of Arithmetical Rings with One Minimal Prime Ideal

Continuing the study of divisibility theory of arithmetical rings started in [1 Ánh, P. N., Márki, L., Vámos, P. (2012). Divisibility theory in commutative rings: Bezout monoidss. Trans. Amer. Math. Soc. 364:3967–3992.[Crossref], [Web of Science ®] , [Google Scholar]] and [2 Ánh, P. N., Siddoway, M. (2010). Divisibility theory of semi-hereditary rings. Proc. Amer. Math. Soc. 138:4231–4242.[Crossref], [Web of Science ®] , [Google Scholar]], we show that the divisibility theory of arithmetical rings with one minimal prime ideal is axiomatizable as Bezout monoids with one minimal m-prime filter. In particular, every Bezout monoid with one minimal m-prime filter is order-isomorphic to the partially ordered monoid with respect to inverse inclusion, of principal ideals in a Bezout ring with a smallest prime ideal. Although this result can be considered as a satisfactory answer to the divisibility theory of both semihereditary domains and valuation rings, the general representation theory of Bezout monoids is still open.     

Generalizations of Prime Ideals

Let R be a commutative ring with identity. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R is weakly prime (resp., almost prime) if a, b ∈ R with ab ∈ I ? {0} (resp., ab ∈ I ? I ²) implies a ∈ I or b ∈ I. Let φ:?(R) → ?(R) ∪ {?} be a function where ?(R) is the set of ideals of R. We call a proper ideal I of R a φ-prime ideal if a, b ∈ R with ab ∈ I ? φ(I) implies a ∈ I or b ∈ I. So taking φ_?(J) = ? (resp., φ₀(J) = 0, φ₂(J) = J ²), a φ_?-prime ideal (resp., φ₀-prime ideal, φ₂-prime ideal) is a prime ideal (resp., weakly prime ideal, almost prime ideal). We show that φ-prime ideals enjoy analogs of many of the properties of prime ideals.     

Prime Ideals of Cancellative Semigroups

Abstract

For a class of prime ideals P of a cancellative semigroup S it is shown that the factors S/P have a structure of a monomial semigroup over a group. Consequences for the semigroup algebras K[S] are discussed.     

The Pointwise Characterizations of Fuzzy Ideals and Fuzzy Prime Ideals

TheconceptofthefuzzysubgroupsintroducedinRosenfeld[l],wasappliedtogeneralizesomeofthebasicconceptsofalgebra.In[2,3,4]LuWangjindefinedfuzzysubringsandfuzzyideals,introducedtheoperationsonfuzzyidea1s.In[5]theautherdiscussedaboutthenatureoffuzzysubringsbythenestedsubring.Butthepointwisedefinitionsoffuzzysubgroupsandfuzzysubringsarerare.Inthispaper,theautherprogramsanddiscussesthefuzzysubrings,fuzzyidealsandfuzzyprimeidealsbyfuzzyPoints..5l.PreliminariesLetusrecallsomeconceptsoccurrjnginthe[2,3…     

Realizing Sets of Associated Prime Ideals

Let R be a Noetherian ring, and let S be a finite subset of Spec R. We characterize when there exists an ideal I such that S equals the set of associated prime divisors of I.     

On Prime Spectrums of Commutative Rings

We show that π-regular rings and clean rings can be completely characterized by topological properties of their prime spectrums respectively. In addition, we give some applications of those result. Among others, we improve the main result of Samei (2004 Samei , K. ( 2004 ). Clean elements in commutative reduced rings . Comm. Algebra 32 ( 9 ): 3479 – 3486 . [CSA] [CROSSREF] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and give a new criterion for a clean ring that a commutative ring is clean if and only if idempotents lifts modulo every radical ideal.     

具有性质（P）的环的同调维数

主要研究具有性质（P）环的同调维数,所得的结果推广了[1]的结果。     

格蕴涵代数的素对偶理想

在格蕴涵代数中定义了一类特殊的素对偶理想,讨论了它们的结构和性质,证明了该格蕴涵代数中的蕴涵运算可以由这些特殊的素对偶理想所确定,并且这些特殊的素对偶理想全体自然地构成一个格蕴涵代数,它和原格蕴涵代数具有格蕴涵代数同构关系。     

Poisson Prime Ideals of Poisson Polynomial Rings

Let A be a finitely generated Poisson algebra over a field of characteristic zero. Here we prove that every Poisson prime ideal of A is prime and give a method to find all Poisson prime ideals in an arbitrary Poisson polynomial ring A[x; α, δ].     

环R的质Fuzzy理想

本文分析了文［３］中Ｆｕｚｚｙ质理想定义之不足，给出了质Ｆｕｚｚｙ理想的概念，讨论了它的等价刻划及性质。