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1.
This paper deals with well-known extensions of the Prüfer domain concept to arbitrary commutative rings. We investigate the transfer of these notions in trivial ring extensions (also called idealizations) of commutative rings by modules and then generate original families of rings with zero-divisors subject to various Prüfer conditions. The new examples give further evidence for the validity of the Bazzoni-Glaz conjecture on the weak global dimension of Gaussian rings. Moreover, trivial ring extensions allow us to widen the scope of validity of Kaplansky-Tsang conjecture on the content ideal of Gaussian polynomials. 相似文献
2.
Sangjib Kim 《Journal of Pure and Applied Algebra》2018,222(2):368-381
The double Pieri algebra, constructed by Howe, Lee, and the author in [10], [12], encodes information on the decomposition of Pieri type tensor products of irreducible representations for complex classical groups. In this paper we study its finite presentation in terms of generators and relations, and then prove that it admits the structure of a cluster algebra. We also study its -module structure. 相似文献
3.
David F. Anderson 《Journal of Pure and Applied Algebra》2007,208(1):351-359
Let D be an integral domain. A saturated multiplicative subset S of D is an almost splitting set if, for each 0≠d∈D, there exists a positive integer n=n(d) such that dn=st for some s∈S and t∈D which is v-coprime to each element of S. We show that every upper to zero in D[X] contains a primary element if and only if D?{0} is an almost splitting set in D[X], if and only if D is a UMT-domain and Cl(D[X]) is torsion. We also prove that D[X] is an almost GCD-domain if and only if D is an almost GCD-domain and Cl(D[X]) is torsion. Using this result, we construct an integral domain D such that Cl(D) is torsion, but Cl(D[X]) is not torsion. 相似文献
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Marco D’Anna Carmelo A. Finocchiaro Marco Fontana 《Journal of Pure and Applied Algebra》2010,214(9):1633-1641
Let f:A→B be a ring homomorphism and let J be an ideal of B. In this paper, we study the amalgamation of A with B along J with respect to f (denoted by A?fJ), a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced and studied by D’Anna and Fontana in 2007, and other classical constructions (such as the A+XB[X], the A+XB?X? and the D+M constructions). In particular, we completely describe the prime spectrum of the amalgamated duplication and we give bounds for its Krull dimension. 相似文献
6.
Mi Hee Park Byung Gyun Kang Phan Thanh Toan 《Journal of Pure and Applied Algebra》2018,222(8):2299-2309
Let be the power series ring over a commutative ring R with identity. For , let denote the content ideal of f, i.e., the ideal of R generated by the coefficients of f. We show that if R is a Prüfer domain and if such that is locally finitely generated (or equivalently locally principal), then a Dedekind–Mertens type formula holds for g, namely for all . More generally for a Prüfer domain R, we prove the content formula for all . As a consequence it is shown that an integral domain R is completely integrally closed if and only if for all nonzero , which is a beautiful result corresponding to the well-known fact that an integral domain R is integrally closed if and only if for all nonzero , where is the polynomial ring over R.For a ring R and , if is not locally finitely generated, then there may be no positive integer k such that for all . Assuming that the locally minimal number of generators of is , Epstein and Shapiro posed a question about the validation of the formula for all . We give a negative answer to this question and show that the finiteness of the locally minimal number of special generators of is in fact a more suitable assumption. More precisely we prove that if the locally minimal number of special generators of is , then for all . As a consequence we show that if is finitely generated (in particular if ), then there exists a nonnegative integer k such that for all . 相似文献
7.
It is not known whether the field of fractions of an integral domain with a compatible lattice order has a compatible lattice order that extends the given order on the integral domain. The polynomial ring
over the real numbers
has a natural compatible lattice order, viz, the coordinatewise order . We describe circumstances in which the field of fractions of
has no archimedean lattice order that extends .Received May 2, 2003; accepted in final form June 4, 2004. 相似文献
8.
Let I be a square-free monomial ideal in R=k[x1,…,xn], and consider the sets of associated primes Ass(Is) for all integers s?1. Although it is known that the sets of associated primes of powers of I eventually stabilize, there are few results about the power at which this stabilization occurs (known as the index of stability). We introduce a family of square-free monomial ideals that can be associated to a finite simple graph G that generalizes the cover ideal construction. When G is a tree, we explicitly determine Ass(Is) for all s?1. As consequences, not only can we compute the index of stability, we can also show that this family of ideals has the persistence property. 相似文献
9.
Let denote the t-class group of an integral domain A. P. Samuel has established that if A is a Krull domain then the mapping , is injective and if A is a regular UFD, then , is bijective. Later, L. Claborn extended this result in case A is a regular Noetherian domain. In the first part of this paper we prove that the mapping ; is an injective homomorphism and in case of an integral domain A such that each υ-invertible υ-ideal of A has υ-finite type, we give an equivalent condition for , to be bijective, thus generalizing the result of Claborn. In the second part of this paper, we define the S-class group of an integral domain A: let S be a (not necessarily saturated) multiplicative subset of an integral domain A. Following [11], a nonzero fractional ideal I of A is S-principal if there exist an and such that . The S-class group of A, S- is the group of fractional t-invertible t-ideals of A under t-multiplication modulo its subgroup of S-principal t-invertible t-ideals of A. We generalize some known results developed for the classic contexts of Krull and PυMD domain and we investigate the case of isomorphism S--. 相似文献
10.
Noômen Jarboui 《Archiv der Mathematik》2008,90(2):133-135
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
11.
Joseph P. S. Kung 《Annals of Combinatorics》1997,1(1):159-172
Just as matroids abstract the algebraic properties of determinants in a vector space, Pfaffian structures abstract the algebraic properties of Pfaffians or skew-symmetric determinants in a symplectic space (that is, a vector space with an alternating bilinear form). This is done using an exchange-augmentation axiom which is a combinatorial version of a Laplace expansion or straightening identity for Pfaffians. Using Pfaffian structures, we study a symplectic analogue of the classical critical problem: given a setS of non-zero vectors in a non-singular symplectic spaceV of dimension2m, find its symplectic critical exponent, that is, the minimum of the set {m?dim(U):U∩S=0}, whereU ranges over all the (totally) isotropic subspaces disjoint fromS. In particular, we derive a formula for the number of isotropic subspaces of a given dimension disjoint from the setS by Möbius inversion over the order ideal of isotropic flats in the lattice of flats of the matroid onS given by linear dependence. This formula implies that the symplectic critical exponent ofS depends only on its matroid and Pfaffian structure; however, it may depend on the dimension of the symplectic spaceV. 相似文献
12.
One of the most frequently referenced monographs on power series rings, “Power Series over Commutative Rings” by James W. Brewer, states in Theorem 21 that if M is a non-SFT maximal ideal of a commutative ring R with identity, then there exists an infinite ascending chain of prime ideals in the power series ring , such that for each n. Moreover, the height of is infinite. In this paper, we show that the above theorem is false by presenting two counter examples. The first counter example shows that the height of can be zero (and hence there is no chain of prime ideals in satisfying for each n). In this example, the ring R is one-dimensional. In the second counter example, we prove that even if the height of is uncountably infinite, there may be no infinite chain of prime ideals in satisfying for each n. 相似文献
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Ahmed Ayache 《Journal of Pure and Applied Algebra》2008,212(1):140-146
The main purpose of this paper is to establish a result giving the number of intermediary rings between R and S when (R,S) is a normal pair of rings and to provide an algorithm to compute this number. 相似文献
15.
We investigate the linear system of thermoelasticity, consisting of an elasticity equation and a heat conduction equation, in a waveguide Ω=(0,1)×Rn−1, with certain boundary conditions. We consider the cases of homogeneous and inhomogeneous systems and prove decay estimates of the solutions, which are a key ingredient to showing the global existence of solutions to non-linear thermoelasticity, after having decomposed the solutions into various parts. We also give a simplified proof to the representation of the solutions to the Cauchy problem of thermoelasticity. 相似文献
16.
Mohamed Jaouhar Ben Abdallah 《Journal of Pure and Applied Algebra》2008,212(10):2170-2175
If 1≤n<∞ and R⊆S 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 R⊆T⊆S) 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
17.
Ahmed Ayache 《Journal of Pure and Applied Algebra》2008,212(10):2176-2181
We establish in this paper a result that gives the number of intermediary rings between R and S where (R,S) is a normal pair of rings. This result answers in particular a question which was left open in [A. Jaballah, Finiteness of the set of intermediary rings in a normal pair, Saintama Math. J. 17 (1999) 59-61]. Further applications are also given. 相似文献
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We explain the construction of fields of formal infinite series in several variables, generalizing the classical notion of formal Laurent series in one variable. Our discussion addresses the field operations for these series (addition, multiplication, and division), the composition, and includes an implicit function theorem. 相似文献