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1.
If 1≤n< and RS are integral domains, then (R,S) is called an n-catenarian pair if for each intermediate ring T (that is each ring T such that RTS) the polynomial ring in n indeterminates, T[n] is catenarian. This implies that (R,S) is m-catenarian for all m<n. The main purpose of this paper is to prove that 1-catenarian and universally catenarian pairs are equivalent in several cases. An example of a 1-catenarian pair which is not 2-catenarian is given.  相似文献   

2.
SoientAB des anneaux (commutaifs et unitaires). On dit que (A,B) est une paire d’anneaux de going-down siD est un anneau de going-down pour tout anneauD tel queADB. On preuve que (A,B) est une paire d’anneaux de going-down si et seulement siA[b 1,b 2] est un anneau de going-down pour toutb 1,b 2 εB.  相似文献   

3.
Wlofgang Müller 《代数通讯》2013,41(8):2687-2695
ABSTRACT:

Let R be a zero-dimensional SFT-ring. It is proved that the minimal prime ideals of the formal power series ring A=R[[x 1, …, xn ]] are the ideals of the form [[x 1, …, xn ]], where is a (minimal) prime of R. It follows that A has Krull dimension n and is catenarian. If R?T where T is also a zero-dimensional SFT-ring, the lying-over, going-up, incomparable, and going-down properties are studied for the extension A?T[[x 1, …, xn ]].  相似文献   

4.
We introduce a new variant of tight closure and give an interpretation of adjoint ideals via this tight closure. As a corollary, we prove that a log pair (X, Δ) is plt if and only if the modulo p reduction of (X, Δ) is divisorially F-regular for all large p ≫ 0. Here, divisorially F-regular pairs are a class of singularities in positive characteristic introduced by Hara and Watanabe (J Algebra Geom 11:363–392, 2002) in terms of Frobenius splitting. The author was partially supported by Grant-in-Aid for Young Scientists (B) 17740021 from JSPS.  相似文献   

5.
《代数通讯》2013,41(6):2489-2500
Elements of the universal (von Neumann) regular ring T(R) of a commutative semiprime ring R can be expressed as a sum of products of elements of R and quasi-inverses of elements of R. The maximum number of terms required is called the regularity degree, an invariant for R measuring how R sits in T(R). It is bounded below by 1 plus the Krull dimension of R. For rings with finitely many primes and integral extensions of noetherian rings of dimension 1, this number is precisely the regularity degree.

For each n ≥ 1, one can find a ring of regularity degree n + 1. This shows that an infinite product of epimorphisms in the category of commutative rings need not be an epimorphism.

Finite upper bounds for the regularity degree are found for noetherian rings R of finite dimension using the Wiegand dimension theory for Patch R. These bounds apply to integral extensions of such rings as well.  相似文献   

6.
We answer a question raised by Othman Echi: Is an E 1 (resp., a C 1) ring an E (resp., a C) ring? We construct a C 1 (thus E 1) ring which is not an E 2 (thus not a C 2) ring. Received: 11 June 2007  相似文献   

7.
David E. Dobbs 《代数通讯》2013,41(6):2603-2623
An integer n is called catenarian if, whenever L/K is an n-dimensional field extension, all maximal chains of fields going from K to L have the same length. Catenarian field extensions and catenarian groups are defined analogously. If n is an even positive integer, 6n is non-catenarian. If n ≥ 3 is odd, there exist infinitely many odd primes p such that p 2 n is non-catenarian. A finite-dimensional field extension is catenarian iff its maximal separable subextension is. If q < p are odd primes where q divides p ? 1 (resp., q divides p + 1), every (resp., not every) group of order p 2 q is catenarian.  相似文献   

8.
David E. Dobbs 《代数通讯》2013,41(10):3553-3572
Many results on going-down domains and divided domains are generalized to the context of rings with von Neumann regular total quotient rings. A (commutative unital) ring R is called regular divided if each P ∈ Spec(R)?(Max(R) ∩ Min(R)) is comparable with each principal regular ideal of R. Among rings having von Neumann regular total quotient rings, the regular divided rings are the pullbacks K× K/P D where K is von Neumann regular, P ∈ Spec(K) and D is a divided domain. Any regular divided ring (for instance, regular comparable ring) with a von Neumann regular total quotient ring is a weak Baer going-down ring. If R is a weak Baer going-down ring and T is an extension ring with a von Neumann regular total quotient ring such that no regular element of R becomes a zero-divisor in T, then R ? T satisfies going-down. If R is a weak Baer ring and P ∈ Spec(R), then R + PR (P) is a going-down ring if and only if R/P and R P are going-down rings. The weak Baer going-down rings R such that Spec(R)?Min(R) has a unique maximal element are characterized in terms of the existence of suitable regular divided overrings.  相似文献   

9.
10.
We propose to give positive answers to the open questions: is R(X,Y) strong S when R(X) is strong S? is R stably strong S (resp., universally catenary) when R[X] is strong S (resp., catenary)? in case R is obtained by a (T,I,D) construction. The importance of these results is due to the fact that this type of ring is the principal source of counterexamples. Moreover, we give an answer to the open questions: is RX1,…,Xn〉 residually Jaffard (resp., totally Jaffard) when R(X1,…,Xn) is ? We construct a three-dimensional local ring R such that R(X1,…,Xn) is totally Jaffard (and hence, residually Jaffard) whereas RX1,…,Xn〉 is not residually Jaffard (and hence, not totally Jaffard).  相似文献   

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