关于严实Hilbert环

在本文中，严实Ｈｉｌｂｅｒｔ环得到了更进一步的刻划．本文的主要结果是：一个环Ａ是严实Ｈｉｌｂｅｒｔ环，当且仅当多项式环Ａ［Ｘ］的每个实极大理想在Ａ上的局限是Ａ的一个极大理想，当且仅当Ａ是实Ｈｉｌｂｅｒｔ环，且Ａ［Ｘ］的每个实极大理想是极大的．     

关于严实Hilbert环

在本文中，严实Ｈｉｌｂｅｒｔ环得到了更进一步的刻划．本文的主要结果是：一个环Ａ是严实Ｈｉｌｂｅｒｔ环，当且仅当多项式环Ａ［Ｘ］的每个实极大理想在Ａ上的局限是Ａ的一个极大理想，当且仅当Ａ是实Ｈｉｌｂｅｒｔ环，且Ａ［Ｘ］的每个实极大理想是极大的．     

关于F-环的一点注记

一个环称为F环,如果环R中含有一个有限非零元集X,使得对任何非零αR与X之交不空(非零)。如果在上面的假设下,X还在R的中心Z(R)中,则称R为FZ环。关于F环,文[1]、[2]给出了一些结果。本文主要结果是: 1.说明文中定理的充分性不真。文[2]的主要定理是:R为半素F-环,当且仅当R为有限个除环上的方阵环的直和。 2.说明非奇异F-环未必是半单环。     

可除的四元数代数上多项式环的Grobner基

设F是一个特征不等于2的域,A是,上的一个可除代数。本文研究了A上多项式环A[x1,X2,…,xn]中理想是有限生成的,以及它的Grobner基;也表明F[x1,x2,…,xn]中有限子集G是F[x1,x2,…,xn]的Griobner基当且仅当G是A[x1,x2,…,xn]中的Grobner基。     

矩阵尾环的Morphic性质(英文)

本文的主要目的是考虑强Morphic环D上的矩阵尾环R[D]的Morphic性质。本文讨论了类似尾环的一些性质。证明了:R[D]是强左Morphic环当且仅当R[D]是左Morphic环当且仅当D是强左Morphic环。本文还构造了一些例子来说明问题。     

McCoy环的Ore扩张(英文)

本文研究了斜多项式环与微分多项式环的McCoy性质,证明了如果环R是α-compatible和可逆的,那么斜多项式R[x;α]是McCoy环当且仅当环R是McCoy环;同时我们也证明了如果环R是δ-compatible与可逆的,那么微分多项式环R[x;δ]是McCoy环当且仅当环R是McCoy环.因此本文对McCoy环的相关结论进行了推广.     

En中齐次多项式芽生成的有限余维理想的判定和应用

本文研究了奇点理论中有限余维理想的一种判定方法,利用Arnold在θn中得出的结论以及Hilbert零点定理,获得C∞实函数芽环En中由齐次多项式芽生成的有限余维理想的特征和判定方法. 其结果是有实用性和有效性的.     

关于Perfect环的一个注记

对于交换环R,Chase[1]证明:对任意集A,若R^A是射影模,则R是一个Artin环.而对非交换环,有例子说明,此结论不成立.本文讨论了对什么环,当R是射影模时,R是一个Artin环.     

强symmetric环

为了统一交换环和约化环的层表示,Lambek引进了Symmetric环.继续symmetric环的研究,定义引入了强symmetric环的概念,研究它的一些扩张性质.证明环R是强symmetric环当且仅当R[x]是强symmetric环当且仅当R[x;x~(-1)]是强symmetric环.也证明对于右Ore环R的经典右商环Q,R是强symmetric环当且仅当Q是强symmetric环.     

罗朗级数环的主拟Baer性

称环 R为右主拟 Baer环（简称为右p·q.Baer环）,如果 R的任意主右理想的右零化子可由幂等元生成.本文证明了,若环 R满足条件Sl（R）（?）C（R）,则罗朗级数环R[[x,x-1]]是右p.q.Baer环当且仅当R是右p.q.Baer环且R的任意可数多个幂等元在I（R）中有广义join.同时还证明了,R是右p.q.Baer环当且仅当R[x,x-1]是右P.q.Baer环.     

Strongly real hilbert rings and the real jacobson semisimplicity

In this paper, several false results in reference [1] related to rela Hilbert rings and the ral Jacobson semisimplicity are negated by a counterexample. By introducing the notion of a strongly real Hilbert ring, we characterize those rings of which every finitely generated real extension is real jacobson semisimple. Moreover, the so-called strictly real Hilbert rings are considered.     

3-Armendariz环的推广(英文)

引入强3-Armendariz环的概念,研究了它们的性质。给出环R是强3-Armendariz环的充要条件。构造了是强3-Armendariz环但不是幂级数Armendariz环的例子。证明了若环R是约化环,则R[X]/(xn)是强3-Armendariz环,其中(xn)是由xn生成的R[x]的理想。     

Proprietes et applications de la notion de contenu

The notion of content is used to solve certain problems. In the first part, we show that the structural morphism of a content algebra (see the paper of D.E. Rush [22] ) is spectrally open, under mild hypothesis. We show also that a flat module is universally content if and only if it is a Mittag-Leffler’s module in the sense of f2ll . In the second part, using content, we exhibit a kind of localization of a commutative ring A, attached to eyery subset X of Spec(A) i.e. a flat morphism A→ X(A). We can thus show that every quasi-compact, stable under generization subset of a spectra is a spectral image under a flat morphism, in a canonical way. We can also give in certain cases an elementary construction of the maximal flat injective epimor-phism of a ring. Suppose that A is a Noetherian ring and consider pro-perties of Noetherian rings such as factoriality, normality and so on.

Let X be the set of prime ideals of A at which A has the property. If X is stable under generization, the flat morphism A→ X(A) verifies hin general the ring X(A) has lornlly the property and a prime ideal P of A has a prime ideal lying over in X(A) if and only if pthe ring has the property at P.     

The ascending chain condition for real ideas

If A is a ring satisfying the ascending chain condition for real ideals, then this condition is also satisfied by the polynomial ring A[X]. However, an example is given to illustrate that the condition need not to hold in the power series ring A[[X]]. It is also shown that if every real prime ideal is the real ideal generated by finitely many elements, then the ring satisfies the ascending chain condition for real ideals. So, the analogues of Hilbert basis theorem and Cohen's theorem hold for real ideals.     

关于凝聚局部环的正则性

本文证明了极大理想m是有限生成的交换凝聚局部环（R,M）是正则的充分必要条件是m可以由一个正则R-序列生成,推广了文献[1]中相应的结论并给出了一个由正则凝聚局部环构造大量的非正则凝聚局部环的方法．     

Quotient rings of polynomial rings

Let R be a commutative ring with identity. The multiplicatively closed sets U₂={fR[X]: c(f)–1=R}, (U₂)={fU₂: f is regular} and S={fR[X]: c(f)=R} are studied. By considering various equalities between these sets, many characterizations of Noetherian rings are found. In particular, a Noetherian ring R has depth 1 if and only if S=(U₂): and each maximal ideal of a Noetherian ring is regular if and only if U₂=(U₂).The theory of Prüfer v-multiplication rings (PVMR's) is developed for rings with zero divisors. Six equivalent conditions are given to the statement that an additively regular v-ring R is a PVMR.     

Dimension de l’anneau des polynomes a valeurs entieres

Let A be a commutative domain with quotient field K and A_S the ring of integer-valued polynomials thus A_S={f∈K[X]; f(A)⊂A}; we show that the Krull dimension of A_S is such that dim A_S≥dim A[X]-1 and give examples where dim A_S=dim A[X]-1. Considering chains of primes of A_S above a maximal idealm of finite residue field, we give also examples where the length of such a chain can arbitrarily be prescribed (whereas in A[X] the length of such chains is always 1). To provide such examples we consider a pair of domains A⊂B sharing an ideal I such that A/I is finite; we give sufficient conditients to have A_S⊂B[X] and show that in this case dim A_S=dim B[X]. At last, as a generalisation of Noetherian rings of dimension 1, we consider domains with an ideal I such that A/I is finite and a power Iⁿ of I is contained in a proper principal ideal of A; for such domains we show that every prime of A_S above a primem containing I is maximal.