EXTENSIONS OF HOMOGENEOUS SEMILOCAL RINGS I

2014 ◽  
Vol 13 (05) ◽  
pp. 1350157
Author(s):  
SUSAN F. EL-DEKEN
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Jacobson Radical ◽  
Local Rings ◽  
Semilocal Ring ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Formal Power

A ring R with Jacobson radical J(R) is a homogeneous semilocal ring if R/J(R) is simple artinian. In this paper, we study the transfer of the property of being homogeneous semilocal from a ring R to the formal power series ring R[[x]], the skew formal power series ring R[[x, α]] and the Hurwitz series ring HR. The results of the paper generalize those proved for commutative local rings. We also consider finite centralizing extensions proving that if the ring of matrices Mn(R) is a homogeneous semilocal ring, then so is R. More generally, if e is an idempotent of a homogeneous semilocal ring S, then eSe is homogeneous semilocal.

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 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.

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.

Some characterizations of pseudo Cohen–Macaulay modules

2015 ◽  
Vol 14 (10) ◽  
pp. 1550142
Author(s):  
Le Thanh Nhan ◽  
Nong Quoc Chinh
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Polynomial Ring ◽  
Power Series Ring ◽  
Series Ring ◽  
Local Case ◽  
Formal Power Series Ring ◽  
Non Local ◽  
System Of Parameters ◽  
Formal Power

Let (R, 𝔪) be a Noetherian local ring and M a finitely generated R-module with dim M = d. Following Cuong and the first author [N. T. Cuong and L. T. Nhan, J. Algebra267 (2003) 156–177], M is called pseudo Cohen–Macaulay if [Formula: see text] for a system of parameters [Formula: see text] of M, where [Formula: see text]. In this paper, first we improve some known results on pseudo Cohen–Macaulay modules. Then we study the localizations of pseudo Cohen–Macaulay modules in order to introduce the pseudo Cohen–Macaulayness for the non-local case. Finally, we give characterizations for the formal power series ring and the polynomial ring being pseudo Cohen–Macaulay.

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.

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