scholarly journals Topics on symbolic Rees algebras for space monomial curves

1991 ◽  
Vol 124 ◽  
pp. 99-132 ◽  
Author(s):  
Shiro Goto ◽  
Koji Nishida ◽  
Yasuhiro Shimoda
Keyword(s):  
Prime Ideals ◽  
Gorenstein Ring ◽  
Regular Local Ring ◽  
Rees Algebra ◽  
Power Series Ring ◽  
Series Ring ◽  
Rees Algebras ◽  
Monomial Curves ◽  
Formal Power Series Ring ◽  
Formal Power

Let A be a regular local ring of dim A = 3 and p a prime ideal in A of dim A/p = 1. We put Rs(p) = (here t denotes an indeterminate over A) and call it the symbolic Rees algebra of p. With this notation the authors [5, 6] investigated the condition under which the A-algebra Rs(p) is Cohen-Macaulay and gave a criterion for Rs(p) to be a Gorenstein ring in terms of the elements f and g in Huneke’s condition [11, Theorem 3.1] of Rs(p) being Noetherian. They furthermore explored the prime ideals p = p(n1, n2, n3) in the formal power series ring A = k[X, Y, Z] over a field k defining space monomial curves and Z = with GCD(n1, n2, nz) = 1 and proved that Rs(p) are Gorenstein rings for certain prime ideals p = p(n1 n2, n3). In the present research, similarly as in [5, 6], we are interested in the ring-theoretic properties of Rs(p) mainly for p = p(n1 n2) nz) and the results of [5, 6] will play key roles in this paper.

scholarly journals Completions of rings of invariants and their divisor class groups

1978 ◽  
Vol 72 ◽  
pp. 71-82 ◽  
Author(s):  
Phillip Griffith
Keyword(s):  
Power Series ◽  
Maximal Ideal ◽  
Total Degree ◽  
Class Groups ◽  
Divisor Class ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Ring Of Polynomials ◽  
Formal Power

Let k be a field and let A = be a normal graded subring of the full ring of polynomials R = k[X1, · · ·, Xn] (where R always is graded via total degree and A0 = k). R. Fossum and the author [F-G] observed that the completion  at the irrelevant maximal ideal of A is isomorphic to the subring of the formal power series ring R̂ = k[[X1, · ·., Xn]] and, moreover, that  is a ring of invariants of an algebraic group whenever A is.

A Note on z-Ideals and z°-Ideals of the Formal Power Series Rings and Polynomial Rings in an Infinite Set of Indeterminates

2020 ◽  
Vol 27 (03) ◽  
pp. 495-508
Author(s):  
Ahmed Maatallah ◽  
Ali Benhissi
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Cardinal Number ◽  
Prime Ideals ◽  
Infinite Cardinal ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Power Series Rings ◽  
Formal Power ◽  
Infinite Set

Let A be a ring. In this paper we generalize some results introduced by Aliabad and Mohamadian. We give a relation between the z-ideals of A and those of the formal power series rings in an infinite set of indeterminates over A. Consider A[[XΛ]]3 and its subrings A[[XΛ]]1, A[[XΛ]]2, and A[[XΛ]]α, where α is an infinite cardinal number. In fact, a z-ideal of the rings defined above is of the form I + (XΛ)i, where i = 1, 2, 3 or an infinite cardinal number and I is a z-ideal of A. In addition, we prove that the same condition given by Aliabad and Mohamadian can be used to get a relation between the minimal prime ideals of the ring of the formal power series in an infinite set of indeterminates and those of the ring of coefficients. As a natural result, we get a relation between the z°-ideals of the formal power series ring in an infinite set of indeterminates and those of the ring of coefficients.

scholarly journals A connectedness theorem over the spectrum of a formal power series ring

2015 ◽  
Vol 26 (11) ◽  
pp. 1550088 ◽  
Author(s):  
Masayuki Kawakita
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Formal Power

We study the connectedness of the non-klt locus over the spectrum of a formal power series ring. In dimension 3, we prove the existence and normality of the smallest lc center, and apply it to the ACC for minimal log discrepancies greater than 1 on smooth 3-folds.

scholarly journals ON DIVISION OF QUASIANALYTIC FUNCTION GERMS

2013 ◽  
Vol 24 (13) ◽  
pp. 1350111 ◽  
Author(s):  
KRZYSZTOF JAN NOWAK
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Local Ring ◽  
Taylor Series ◽  
Necessary Condition ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Formal Power ◽  
Sufficient And Necessary Condition

We establish the following criterion for divisibility in the local ring [Formula: see text] of those quasianalytic function germs at 0 ∈ ℝn which are definable in a polynomially bounded structure. A sufficient (and necessary) condition for the divisibility of two function germs in [Formula: see text] is that of their Taylor series at 0 ∈ ℝn in the formal power series ring.

Vanishing of Hochschild Cohomologies for Local Rings with Embedding Dimension Two

1995 ◽  
Vol 38 (1) ◽  
pp. 59-65 ◽  
Author(s):  
Mitsuo Hoshino
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Primary Ideal ◽  
Local Rings ◽  
Embedding Dimension ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Formal Power

AbstractLet S = k[[x,y]] be a formal power series ring in two variables x, y over a field k and I an (x, y)-primary ideal of S. We show that S/I is selfinjective if Hi(S/I, S/I ⊗k S/I) = 0 for i = 1 and 2.

scholarly journals Hypertranscendental elements of a formal power-series ring of positive characteristic

1992 ◽  
Vol 125 ◽  
pp. 93-103 ◽  
Author(s):  
Kayoko Shikishima-Tsuji ◽  
Masashi Katsura
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Positive Characteristic ◽  
Rational Numbers ◽  
Natural Numbers ◽  
Real Numbers ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Formal Power

Throughout this paper, we denote by N, Q and R the set of all natural numbers containing 0, the set of all rational numbers, and the set of all real numbers, respectively.

On transfer of annihilator conditions of rings

2018 ◽  
Vol 17 (10) ◽  
pp. 1850199
Author(s):  
Abdollah Alhevaz ◽  
Ebrahim Hashemi ◽  
Rasul Mohammadi
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Noetherian Rings ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Base Ring ◽  
Polynomial Formula ◽  
Formal Power

It is well known that a polynomial [Formula: see text] over a commutative ring [Formula: see text] with identity is a zero-divisor in [Formula: see text] if and only if [Formula: see text] has a non-zero annihilator in the base ring, where [Formula: see text] is the polynomial ring with indeterminate [Formula: see text] over [Formula: see text]. But this result fails in non-commutative rings and in the case of formal power series ring. In this paper, we consider the problem of determining some annihilator properties of the formal power series ring [Formula: see text] over an associative non-commutative ring [Formula: see text]. We investigate relations between power series-wise McCoy property and other standard ring-theoretic properties. In this context, we consider right zip rings, right strongly [Formula: see text] rings and rings with right Property [Formula: see text]. We give a generalization (in the case of non-commutative ring) of a classical results related to the annihilator of formal power series rings over the commutative Noetherian rings. We also give a partial answer, in the case of formal power series ring, to the question posed in [1 Question, p. 16].

scholarly journals A Study of Cyclic Codes Via a Surjective Mapping

MATEMATIKA ◽  
2018 ◽  
Vol 34 (2) ◽  
pp. 325-332
Author(s):  
Mriganka Sekhar Dutta ◽  
Helen K. Saikia
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Cyclic Codes ◽  
Cyclic Shift ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Surjective Mapping ◽  
Formal Power

In this article, cyclic codes of length $n$ over a formal power series ring and cyclic codes of length $nl$ over a finite field are studied. By defining a module isomorphism between $R^n$ and $(Z_4)^{2^kn}$, Dinh and Lopez-Permouth proved that a cyclic shift in $(Z_4)^{2^kn}$ corresponds to a constacyclic shift in $R^n$ by $u$, where $R=\frac{Z_4[u]}{<u^{2^k}-1>}$. We have defined a bijective mapping $\Phi_l$ on $R_{\infty}$, where $R_{\infty}$ is the formal power series ring over a finite field $\mathbb{F}$. We have proved that a cyclic shift in $(\mathbb{F})^{ln}$ corresponds to a $\Phi_l-$cyclic shift in $(R_{\infty})^n$ by defining a mapping from $(R_{\infty})^n$ onto $(\mathbb{F})^{ln}$. We have also derived some related results.

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