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>

The semigroup of endomorphisms of a Boolean ring

1970 ◽  
Vol 11 (4) ◽  
pp. 411-416 ◽  
Author(s):  
Kenneth D. Magill
Keyword(s):  
Rational Numbers ◽  
Boolean Ring ◽  
Ring Of Integers ◽  
The Family ◽  
Identity Automorphism

The family (R) of all endomorphisms of a ring R is a semigroup under composition. It follows easily that if R and T are isomorphic rings, then (R) and (T) are isomorphic semigroups. We devote ourselves here to the converse question: ‘If (R) and (T) are isomorphic, must R and T be isomorphic?’ As one might expect, the answer is, in general, negative. For example, the ring of integers has precisely two endomorphisms – the zero endomorphism and the identity automorphism. Since the same is true of the ring of rational numbers, the two endomorphism semigroups are isomorphic while the rings themselves are certainly not.

scholarly journals Recent results on weakly factorial domains

2018 ◽  
Vol 20 ◽  
pp. 01001
Author(s):  
Chang Gyu Whan
Keyword(s):  
Polynomial Ring ◽  
Quotient Group ◽  
Integral Domain ◽  
Laurent Polynomial ◽  
Semigroup Ring ◽  
Prime Element ◽  
Cancellative Monoid ◽  
Factorial Domain ◽  
Laurent Polynomial Ring

In this paper, we will survey recent results on weakly factorial domains base on the results of [11, 13, 14]. LetD be an integral domain, X be an indeterminate over D, d ∈ D, R = D[X,d/X] be a subring of the Laurent polynomial ring D[X,1/X], Γ be a nonzero torsionless commutative cancellative monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. Among other things, we show that R is a weakly factorial domain if and only if D is a weakly factorial GCD‐domain and d = 0, d is a unit of D or d is a prime element of D. We also show that if char(D) = 0 (resp., char(D) = p > 0), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD domain, Γ is a weakly factorial GCD semigroup, and G is of type (0,0,0,…) (resp., (0,0,0,…) except p).

Recursive elements and constructive extensions of computable local integral domains

1973 ◽  
Vol 38 (2) ◽  
pp. 272-290 ◽  
Author(s):  
Glen H. Suter
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Integral Domain ◽  
Rational Numbers ◽  
Algebraic Structures ◽  
Computable Structures ◽  
Integral Domains ◽  
Local Integral ◽  
Algebraic Operations ◽  
Axiom Systems

With reservations, one can think of abstract algebra as the study of what consequences can be drawn from the axioms associated with certain concrete algebraic structures. Two important examples of such concrete algebraic structures are the integers and the rational numbers. The integers and the rational numbers have two properties which are not in general mirrored in the abstract axiom systems associated with them. That is, the integers and the rational numbers both have effectively computable metrics and their algebraic operations are effectively computable. The study of abstract algebraic systems which possess effectively computable algebraic operations has produced many interesting results. One can think of a computable algebraic structure as one whose elements have been labeled by the set of positive integers and whose operations are effectively computable. The formal definition of computable local integral domain will be given in §3. Some specific computable structures which have been studied are the integers, the rational numbers, and the rational numbers with p-adic valuation. Computable structures were studied in general by Rabin [12]. This paper concerns computable local integral domains as exemplified by the local integral domain Zp. Zp is the localization of the integers with respect to the maximal prime ideal generated by the positive prime p. We should note that the concept of local integral domain is not first order.Let the ordered pair (Q, M) stand for a local ring, where Q is the local ring and M is the unique maximal prime ideal of Q. Since most of my results are proving the existence of certain effective procedures, the assumption that Q has a principal maximal ideal M (rather than M has n generators) greatly simplifies many of the proofs.

scholarly journals An explicit Baker-type lower bound of exponential values

2015 ◽  
Vol 145 (6) ◽  
pp. 1153-1182 ◽  
Author(s):  
Anne-Maria Ernvall-Hytönen ◽  
Kalle Leppälä ◽  
Tapani Matala-aho
Keyword(s):  
Lower Bound ◽  
Linear Form ◽  
Quadratic Field ◽  
Rational Numbers ◽  
Rational Points ◽  
Imaginary Quadratic Field ◽  
Ring Of Integers

Let 𝕀 denote an imaginary quadratic field or the field ℚ of rational numbers and let ℤ𝕀denote its ring of integers. We shall prove a new explicit Baker-type lower bound for a ℤ𝕀-linear form in the numbers 1, eα1, . . . , eαm,m⩾ 2, whereα0= 0,α1, . . . ,αmarem+ 1 different numbers from the field 𝕀. Our work gives substantial improvements on the existing explicit versions of Baker’s work about exponential values at rational points. In particular, dependencies onmare improved.

GAUSS AND KLOOSTERMAN SUMS OVER RESIDUE RINGS OF ALGEBRAIC INTEGERS

2011 ◽  
Vol 07 (01) ◽  
pp. 101-114
Author(s):  
S. GURAK
Keyword(s):  
Classical Result ◽  
Classical Case ◽  
Rational Numbers ◽  
Kloosterman Sums ◽  
Gauss Sum ◽  
Algebraic Integers ◽  
Ring Of Integers ◽  
Rings Of Algebraic Integers ◽  
Sum Formula ◽  
Recent Evaluation

Let K be a field of degree n over Q, the field of rational numbers, with ring of integers O. Fix an integer m > 1, say with [Formula: see text] as a product of distinct prime powers, and let χ be a numerical character modulo m of conductor f(χ). Set ζm = exp (2πi/m) and let M be any ideal of O satisfying Tr M ⊆ mZ and N(1 + M) ⊆ 1 + f(χ)Z, where Tr and N are the trace and norm maps for K/Q. Then the Gauss sum [Formula: see text] is well-defined. If in addition N(1 + M) ⊆ 1 + mZ, then the Kloosterman sums [Formula: see text] are well-defined for any numerical character η ( mod m). The computation of GM(χ) and RM(η, z) is shown to reduce to their determination for m = pr, a power of a prime p, where M is comprised solely of ideals of K lying above p. In this setting we first explicitly determine GM(χ) for m = pr (r > 1) generalizing Mauclaire's classical result for K = Q. Relying on the recent evaluation of Kloosterman sums for prime powers in p-adic fields, we then proceed to compute the Kloosterman sums RM(η, z) here for m = pr (r > 1) when o(η) | p -1. This determination generalizes Salie's result in the classical case K = Q with o(η) = 1 or 2.

Two Unusual Representations for the Set of Real Numbers

1970 ◽  
Vol 63 (8) ◽  
pp. 665
Author(s):  
Sanderson M. Smith
Keyword(s):  
Rational Numbers ◽  
Real Numbers ◽  
The Real ◽  
Binary Operations

The rational numbers and the real numbers are both fields under the binary operations of addition and multiplication.

ON MAXIMAL NON-UNIVERSALLY CATENARIAN SUBRINGS

2008 ◽  
Vol 07 (05) ◽  
pp. 553-556 ◽  
Author(s):  
MABROUK BEN NASR ◽  
NOÔMEN JARBOUI
Keyword(s):  
Polynomial Ring ◽  
Integral Domain ◽  
Intermediate Ring

An integral domain R with field of fractions K is called a maximal non-1-catenarian subring of K if the polynomial ring in one variable, R[X] is not catenarian and for each proper intermediate ring T (that is each ring T such that R ⊂ T ⊆ K) T[X] is catenarian. The main purpose of this paper is to prove that the concept of maximal non-1-catenarian subrings and that of maximal non-universally catenarian subrings are equivalent.

On Duo Group Rings

2011 ◽  
Vol 18 (01) ◽  
pp. 163-170
Author(s):  
Weidong Gao ◽  
Yuanlin Li
Keyword(s):  
Group Ring ◽  
Characteristic Zero ◽  
Integral Domain ◽  
Torsion Group ◽  
Rational Numbers ◽  
Group Rings ◽  
Quaternion Group ◽  
Quotient Field ◽  
Ring Of Algebraic Integers ◽  
Field Integral

It is shown that if the group ring RQ8 of the quaternion group Q8 of order 8 over an integral domain R is duo, then R is a field for the following cases: (1) char R ≠ 0, and (2) char R = 0 and S ⊆ R ⊆ KS, where S is a ring of algebraic integers and KS is its quotient field. Hence, we confirm that the field ℚ of rational numbers is the smallest integral domain R of characteristic zero such that RQ8 is duo. A non-field integral domain R of characteristic zero for which RQ8 is duo is also identified. Moreover, we give a description of when the group ring RG of a torsion group G is duo.

Integers of Biquadratic Fields

1970 ◽  
Vol 13 (4) ◽  
pp. 519-526 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Explicit Form ◽  
Integral Domain ◽  
Minimal Polynomial ◽  
Rational Numbers ◽  
Integral Basis ◽  
Squarefree Integers ◽  
Biquadratic Fields ◽  
Rational Integral

Let Q denote the field of rational numbers. If m, n are distinct squarefree integers the field formed by adjoining √m and √n to Q is denoted by Q(√m, √n). Since Q(√m, √n) = Q(√m, √n) and √m + √n has for its unique minimal polynomial x4 —2(m + n)x2 + (m - n)2, Q(√m, √n) is a biquadratic field over Q. The elements of Q(√m, √n) are of the form a0 + a1√m + a2√n + a3√mn, where a1, a2, a3 ∊ Q. Any element of Q(√m, √n) which satisfies a monic equation of degree ≥ 1 with rational integral coefficients is called an integer of Q(√m, √n). The set of all these integers is an integral domain. In this paper we determine the explicit form of the integers of Q(√m, √n) (Theorem 1), an integral basis for Q(√m, √n) (Theorem 2), and the discriminant of Q(√m, √n) (Theorem 3).

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