Divisibility properties of the quotient ring of the polynomial ring D[X, Y, U, V] modulo (XV – YU)

2021 ◽  
Author(s):  
Gyu Whan Chang
Keyword(s):  
Polynomial Ring ◽  
Quotient Ring ◽  
Divisibility Properties

scholarly journals The canonical modules of graded rings defined by generic matrices

1981 ◽  
Vol 81 ◽  
pp. 105-112 ◽  
Author(s):  
Yuji Yoshino
Keyword(s):  
Polynomial Ring ◽  
Quotient Ring ◽  
Graded Rings ◽  
Canonical Module ◽  
Graded Ring ◽  
The Ideal

Let k be a field, and X = [xij] be an n × (n + m) matrix whose elements are algebraically independent over k.We shall study the canonical module of the graded ring R, which is a quotient ring of the polynomial ring A = k[X] by the ideal αn(X) generated by all the n × n minors of X.

SKEW POLYNOMIAL RINGS WHICH ARE GENERALIZED ASANO PRIME RINGS

2013 ◽  
Vol 12 (07) ◽  
pp. 1350024 ◽  
Author(s):  
H. MARUBAYASHI ◽  
INTAN MUCHTADI-ALAMSYAH ◽  
A. UEDA
Keyword(s):  
Polynomial Ring ◽  
Prime Ring ◽  
Quotient Ring ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Prime Rings ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

Let R be a prime Goldie ring with quotient ring Q and σ be an automorphism of R. We define (σ-) generalized Asano prime rings and prove that a skew polynomial ring R[x; σ] is a generalized Asano prime ring if and only if R is a σ-generalized Asano prime ring. This is done by giving explicitly the structure of all v-ideals of R[x; σ] in case R is a σ-Krull prime ring. We provide some examples of σ-generalized Asano prime rings which are not Krull prime rings.

IDEAL COSET INVARIANTS FOR SURFACE-LINKS IN ℝ4

2013 ◽  
Vol 22 (09) ◽  
pp. 1350052 ◽  
Author(s):  
YEWON JOUNG ◽  
JIEON KIM ◽  
SANG YOUL LEE
Keyword(s):  
Normal Form ◽  
Polynomial Ring ◽  
Gröbner Basis ◽  
Quotient Ring ◽  
Grobner Basis ◽  
Link Invariant ◽  
Link Invariants ◽  
Classical Link ◽  
Marked Graph

In [Towards invariants of surfaces in 4-space via classical link invariants, Trans. Amer. Math. Soc.361 (2009) 237–265], Lee defined a polynomial [[D]] for marked graph diagrams D of surface-links in 4-space by using a state-sum model involving a given classical link invariant. In this paper, we deal with some obstructions to obtain an invariant for surface-links represented by marked graph diagrams D by using the polynomial [[D]] and introduce an ideal coset invariant for surface-links, which is defined to be the coset of the polynomial [[D]] in a quotient ring of a certain polynomial ring modulo some ideal and represented by a unique normal form, i.e. a unique representative for the coset of [[D]] that can be calculated from [[D]] with the help of a Gröbner basis package on computer.

The Erdős–Burgess constant of the multiplicative semigroup of a factor ring of 𝔽q[x]

2019 ◽  
Vol 15 (01) ◽  
pp. 131-136 ◽  
Author(s):  
Haoli Wang ◽  
Jun Hao ◽  
Lizhen Zhang
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Prime Ideal ◽  
Polynomial Ring ◽  
Prime Power ◽  
Quotient Ring ◽  
Prime Ideals ◽  
Multiplicative Semigroup ◽  
Associative Operation ◽  
Sharp Lower Bound

Let [Formula: see text] be a commutative semigroup endowed with a binary associative operation [Formula: see text]. An element [Formula: see text] of [Formula: see text] is said to be idempotent if [Formula: see text]. The Erdős–Burgess constant of [Formula: see text] is defined as the smallest [Formula: see text] such that any sequence [Formula: see text] of terms from [Formula: see text] and of length [Formula: see text] contains a nonempty subsequence, the sum of whose terms is idempotent. Let [Formula: see text] be a prime power, and let [Formula: see text] be the polynomial ring over the finite field [Formula: see text]. Let [Formula: see text] be a quotient ring of [Formula: see text] modulo any ideal [Formula: see text]. We gave a sharp lower bound of the Erdős–Burgess constant of the multiplicative semigroup of the ring [Formula: see text], in particular, we determined the Erdős–Burgess constant in the case when [Formula: see text] is the power of a prime ideal or a product of pairwise distinct prime ideals in [Formula: see text].

scholarly journals Elliptic curve over a finite ring generated by 1 and an idempotent element $e$ with coefficients in the finite field $\mathbb{F}_{3^d}$

2021 ◽  
Vol 40 ◽  
pp. 1-19
Author(s):  
A. Boulbot ◽  
Abdelhakim Chillali ◽  
A Mouhib
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Elliptic Curve ◽  
Polynomial Ring ◽  
Quotient Ring ◽  
Finite Ring ◽  
Idempotent Element ◽  
Characteristic 3 ◽  
Specific Equation ◽  
Weierstrass Equation

An elliptic curve over a ring $\mathcal{R}$ is a curve in the projective plane $\mathbb{P}^{2}(\mathcal{R})$ given by a specific equation of the form $f(X, Y, Z)=0$ named the Weierstrass equation, where $f(X, Y, Z)=Y^2Z+a_1XYZ+a_3YZ^2-X^3-a_2X^2Z-a_4XZ^2-a_6Z^3$ with coefficients $a_1, a_2, a_3, a_4, a_6$ in $\mathcal{R}$ and with an invertible discriminant in the ring $\mathcal{R}.$ %(see \cite[Chapter III, Section 1]{sil1}).  In this paper, we consider an elliptic curve over a finite ring of characteristic 3 given by the Weierstrass equation: $Y^2Z=X^3+aX^2Z+bZ^3$ where $a$ and $b$ are in the quotient ring $\mathcal{R}:=\mathbb{F}_{3^d}[X]/(X^2-X),$ where $d$ is a positive integer and $\mathbb{F}_{3^d}[X]$ is the polynomial ring with coefficients in the finite field $\mathbb{F}_{3^d}$ and such that $-a^3b$ is invertible in $\mathcal{R}$.

scholarly journals Weakly Closed Graphs and F-Purity of Binomial Edge Ideals

2018 ◽  
Vol 25 (04) ◽  
pp. 567-578
Author(s):  
Kazunori Matsuda
Keyword(s):  
Polynomial Ring ◽  
Quotient Ring ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Edge Ideal ◽  
Closed Graph ◽  
Edge Ideals ◽  
Weakly Closed

Herzog, Hibi, Hreindóttir et al. introduced the class of closed graphs, and they proved that the binomial edge ideal JG of a graph G has quadratic Gröbner bases if G is closed. In this paper, we introduce the class of weakly closed graphs as a generalization of the closed graph, and we prove that the quotient ring S/JG of the polynomial ring [Formula: see text] with K a field and [Formula: see text] is F-pure if G is weakly closed. This fact is a generalization of Ohtani’s theorem.

Note on the Davenport constant of the multiplicative semigroup of the quotient ring 𝔽p[x] 〈f(x)〉

2016 ◽  
Vol 12 (03) ◽  
pp. 663-669 ◽  
Author(s):  
Haoli Wang ◽  
Lizhen Zhang ◽  
Qinghong Wang ◽  
Yongke Qu
Keyword(s):  
Positive Integer ◽  
Commutative Semigroup ◽  
Polynomial Ring ◽  
Quotient Ring ◽  
Multiplicative Semigroup ◽  
Davenport Constant ◽  
Prime Field ◽  
Group Of Units ◽  
Polynomial Formula ◽  
Finite Commutative Semigroup

Let [Formula: see text] be a finite commutative semigroup. The Davenport constant of [Formula: see text], denoted [Formula: see text], is defined to be the least positive integer [Formula: see text] such that every sequence [Formula: see text] of elements in [Formula: see text] of length at least [Formula: see text] contains a proper subsequence [Formula: see text] with the sum of all terms from [Formula: see text] equaling the sum of all terms from [Formula: see text]. Let [Formula: see text] be a polynomial ring in one variable over the prime field [Formula: see text], and let [Formula: see text]. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the quotient ring [Formula: see text] and proved that, for any prime [Formula: see text] and any polynomial [Formula: see text] which factors into a product of pairwise non-associate irreducible polynomials, [Formula: see text] where [Formula: see text] denotes the multiplicative semigroup of the quotient ring [Formula: see text] and [Formula: see text] denotes the group of units of the semigroup [Formula: see text].

scholarly journals MORPHIC PROPERTY OF A QUOTIENT RING OVER POLYNOMIAL RING

2013 ◽  
Vol 50 (5) ◽  
pp. 1433-1439
Author(s):  
Kai Long ◽  
Qichuan Wang ◽  
Lianggui Feng
Keyword(s):  
Polynomial Ring ◽  
Quotient Ring

Pembentukan Lapangan Faktor dari Suatu Daerah Integral

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Tri Widjajanti ◽  
Dahlia Ramlan ◽  
Rium Hilum
Keyword(s):  
Polynomial Ring ◽  
Integral Domain ◽  
Rational Numbers ◽  
Ring Of Integers ◽  
Binary Operations

<em>Ring of integers under the addition and multiplication as integral domain can be imbedded to the field of rational numbers. In this paper we make&nbsp; a construction such that any integral domain can be&nbsp; a field of quotient. The construction contains three steps. First, we define element of field F from elements of integral domain D. Secondly, we show that the binary operations in fare well-defined. Finally, we prove that </em><em>&nbsp;</em><em>f</em><em> </em><em>:</em><em> </em><em>D </em><em>&reg;</em><em> </em><em>F is an isomorphisma. In this case, the polynomial ring F[x] as the integral domain can be imbedded to the field of quotient.</em>

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