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1.
Louis H. Rowen 《代数通讯》2013,41(6):2263-2279
We consider maximal left ideals L of the polynomial ring R[λ1, …, λn], for R noncommutative. In §1 we reprove and generalize Resco's result that any maximal left ideal L is generated by ≤ n elements whenever R is simple Artinian, and obtain more precise information about the generators when R satisfies a PI. In many instances, fewer than n generators suffice; this is considered in §3, by means of various examples. In §2 we see by a straightforward argument that L has bounded height as a prime left ideal whenever R is a simple Pl-ring, but this does not- hold in general for R simple Artinian.  相似文献   

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Summary Study of relations between the prime and maximal spectra of a ringA and ofA[X], without noetherian assumptions. Application to the cases whereA has finite noetherian type andA is an arbitrary valuation domain; behaviour of the catenary property. New proofs of known results aboutG-ideals and Hilbert domains.
Riassunto Si studiano le relazioni fra lo spettro ideale e quello massimale di un anelloA e diA[X] senza ipotesi di noetherianità. Si fanno delle applicazioni ai casi in cuiA è un anello di tipo noetheriano finito o è un arbitrario dominio di valutazione; si studia inoltre il comportamento della proprietà catenaria. Si danno nuove dimostrazioni di risultati noti suG-ideali e domini di Hilbert.
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An idealI of the ringK[x 1, ...,x n ] of polynomials over a fieldK inn indeterminates is a full ideal ifI is closed under substitution,f I,g 1...gn K[x 1, ...,x n ] implyf(g 1, ...,g n ) I. In this paper we continue the investigation of full ideals ofK[x 1, ...,x n ]. In particular we determine several classes of full ideals ofK[x, y] (K a finite field) and investigate properties of these classes.The first author gratefully acknowledges support from theDeutsche Forschungsgemeinschaft  相似文献   

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We extend a theorem of Kist for commutative PP rings to principally quasi-Baer rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. Also decompositions of quasi-Baer and principally quasi-Baer rings are investigated. This revised version was published online in June 2006 with corrections to the Cover Date.  相似文献   

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In this paper restricted differential operator rings are studied. A restricted differential operator ring is an extension of ak-algebraR by the restricted enveloping algebra of a restricted Lie algebra g which acts onR. This is an example of a smash productR #H whereH=u (g). We actually deal with a more general twisted construction denoted byR * g where the restricted Lie algebra g is not necessarily embedded isomorphically inR * g. Assume that g is finite dimensional abelian. The principal result obtained is Incomparability, which states that prime idealsP 1P 2R * g have different intersections withR. We also study minimal prime ideals ofR * g whenR is g-prime, showing that the minimal primes are precisely those having trivial intersection withR, that these primes are finite in number, and their intersection is a nilpotent ideal. Prime and primitive ranks are considered as an application of the foregoing results.  相似文献   

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We call an ideal I of a commutative ring R radically perfect if among the ideals of R 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 Noetherian integral domain of Krull dimension one containing a field of characteristic zero. Then each prime ideal of the polynomial ring R[X] is radically perfect if and only if R is a Dedekind domain with torsion ideal class group. We also show that over a finite dimensional Bézout domain R, the polynomial ring R[X] has the property that each prime ideal of it is radically perfect if and only if R is of dimension one and each prime ideal of R is the radical of a principal ideal.  相似文献   

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Let R be an hereditary Noetherian prime ring (or, HNP-ring, for short), and let S?=?R[x;σ] be a skew polynomial ring over R with σ being an automorphism on R. The aim of the paper is to describe completely the structure of right projective ideals of R[x;σ] where R is an HNP-ring and to obtain that any right projective ideal of S is of the form X𝔟[x;σ], where X is an invertible ideal of S and 𝔟 is a σ-invariant eventually idempotent ideal of R.  相似文献   

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Let R be a ring graded by an abelian group.We study prime ideals of R that are maximal for not containing nonzero homogeneous elements.Also prime ideals of the symmetric graded Martindale ring of quotients of R are investigated.The results are applied to study when R is a Jacobson ring in case R is a Z-graded ring or a group ring of a finitely generated abelian group, or in case R is right Noetherian and strongly graded by a polycyclic-by-finite group.  相似文献   

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