Bounds for the solutions of norm form, discriminant form and index form equations in finitely generated integral domains

1983 ◽  
Vol 42 (1-2) ◽  
pp. 45-80 ◽  
Author(s):  
K. Győry
Keyword(s):  
Finitely Generated ◽  
Integral Domains ◽  
Index Form ◽  
Norm Form ◽  
Discriminant Form

scholarly journals Higher derivations on finitely generated integral domains. II

1975 ◽  
Vol 51 (1) ◽  
pp. 8-8 ◽  
Author(s):  
William C. Brown
Keyword(s):  
Finitely Generated ◽  
Integral Domains

scholarly journals Non-isomorphic rings with isomorphic matrix rings

1993 ◽  
Vol 36 (2) ◽  
pp. 339-348 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Finitely Generated ◽  
Integral Domains ◽  
Matrix Rings ◽  
Uncountable Family

We construct an uncountable family of pairwise non-isomorphic rings Si, such that the corresponding full 2 by 2 matrix rings M2(Si) are all isomorphic to each other. The rings Si are Noetherian integral domains which are finitely-generated as modules over their centres.

THE v-OPERATION IN EXTENSIONS OF INTEGRAL DOMAINS

2012 ◽  
Vol 11 (01) ◽  
pp. 1250007 ◽  
Author(s):  
DAVID F. ANDERSON ◽  
SAID EL BAGHDADI ◽  
MUHAMMAD ZAFRULLAH
Keyword(s):  
Finitely Generated ◽  
Integral Domains ◽  
Fractional Ideal

An extension D ⊆ R of integral domains is strongly t-compatible (respectively, t-compatible) if (IR)-1 = (I-1R)v (respectively, (IR)v = (IvR)v) for every nonzero finitely generated fractional ideal I of D. We show that strongly t-compatible implies t-compatible and give examples to show that the converse does not hold. We also indicate situations where strong t-compatibility and its variants show up naturally. In addition, we study integral domains D such that D ⊆ R is strongly t-compatible (respectively, t-compatible) for every overring R of D.

scholarly journals On finitely generated birational flat extensions of integral domains

2004 ◽  
Vol 11 (1) ◽  
pp. 35-40 ◽  
Author(s):  
Susumu Oda
Keyword(s):  
Finitely Generated ◽  
Integral Domains

scholarly journals Effective results for unit equations over finitely generated integral domains

2012 ◽  
Vol 154 (2) ◽  
pp. 351-380 ◽  
Author(s):  
JAN–HENDRIK EVERTSE ◽  
KÁLMÁN GYŐRY
Keyword(s):  
Integral Domain ◽  
Linear Equations ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Polynomial Rings ◽  
Finitely Generated ◽  
Integral Domains ◽  
Finiteness Result ◽  
Unit Equations ◽  
And Function

AbstractLet A ⊃ ℤ be an integral domain which is finitely generated over ℤ and let a,b,c be non-zero elements of A. Extending earlier work of Siegel, Mahler and Parry, in 1960 Lang proved that the equation (*) aϵ +bη = c in ϵ, η ∈ A* has only finitely many solutions. Using Baker's theory of logarithmic forms, Győry proved, in 1979, that the solutions of (*) can be determined effectively if A is contained in an algebraic number field. In this paper we prove, in a quantitative form, an effective finiteness result for equations (*) over an arbitrary integral domain A of characteristic 0 which is finitely generated over ℤ. Our main tools are already existing effective finiteness results for (*) over number fields and function fields, an effective specialization argument developed by Győry in the 1980's, effective results of Hermann (1926) and Seidenberg (1974) on linear equations over polynomial rings over fields, and similar such results by Aschenbrenner, from 2004, on linear equations over polynomial rings over ℤ. We prove also an effective result for the exponential equation aγ1v1···γsvs+bγ1w1 ··· γsws=c in integers v1,…,ws, where a,b,c and γ1,…,γs are non-zero elements of A.

Algorithmic Resolution of Discriminant Form and Index Form Equations

2016 ◽  
pp. 117-147
Author(s):  
Jan-Hendrik Evertse ◽  
Kalman Gyory
Keyword(s):  
Index Form ◽  
Discriminant Form

scholarly journals Extended semi-hereditary rings

1978 ◽  
Vol 26 (4) ◽  
pp. 465-474 ◽  
Author(s):  
M. W. Evans
Keyword(s):  
Finitely Generated ◽  
Integral Domains ◽  
Hereditary Ring ◽  
Prufer Domains ◽  
Prüfer Domains

AbstractA ring R for which every finitely generated right submodule of SR, the left flat epimorphic hull of R, is projective is termed an extended semi-hereditary ring. It is shown that several of the characterizing properties of Prufer domains may be generalized to give characterizations of extended semi-hereditary rings. A suitable class of PP rings is introduced which in this case serves as a generalization of commutative integral domains.

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