Commutative Ideal Theory without Finiteness Conditions: Irreducibility in the Quotient Field

2016 ◽  
pp. 157-182
Keyword(s):  
Ideal Theory ◽  
Quotient Field ◽  
Finiteness Conditions ◽  
Commutative Ideal

scholarly journals Commutative ideal theory without finiteness conditions: Primal ideals

2004 ◽  
Vol 357 (7) ◽  
pp. 2771-2798 ◽  
Author(s):  
Laszlo Fuchs ◽  
William Heinzer ◽  
Bruce Olberding
Keyword(s):  
Ideal Theory ◽  
Finiteness Conditions ◽  
Commutative Ideal

scholarly journals Commutative ideal theory without finiteness conditions: Completely irreducible ideals

2006 ◽  
Vol 358 (7) ◽  
pp. 3113-3131 ◽  
Author(s):  
Laszlo Fuchs ◽  
William Heinzer ◽  
Bruce Olberding
Keyword(s):  
Ideal Theory ◽  
Finiteness Conditions ◽  
Commutative Ideal

Commutative Ideal Theory without Finiteness Conditions

2006 ◽  
pp. 121-145 ◽  
Author(s):  
Laszlo Fuchs ◽  
WilliamHeinzer ◽  
Bruce Olberding
Keyword(s):  
Ideal Theory ◽  
Finiteness Conditions ◽  
Commutative Ideal

scholarly journals Abstract Commutative Ideal Theory

1990 ◽  
pp. 369-386
Author(s):  
R. P. Dilworth
Keyword(s):  
Ideal Theory ◽  
Commutative Ideal

scholarly journals The monotone catenary degree of monoids of ideals

2019 ◽  
Vol 29 (03) ◽  
pp. 419-457 ◽  
Author(s):  
Alfred Geroldinger ◽  
Andreas Reinhart
Keyword(s):  
Ideal Theory ◽  
Integral Domains ◽  
Finiteness Conditions ◽  
Central Topic ◽  
Sets Of Lengths ◽  
Catenary Degree

Factoring ideals in integral domains is a central topic in multiplicative ideal theory. In the present paper, we study monoids of ideals and consider factorizations of ideals into multiplicatively irreducible ideals. The focus is on the monoid of nonzero divisorial ideals and on the monoid of [Formula: see text]-invertible divisorial ideals in weakly Krull Mori domains. Under suitable algebraic finiteness conditions, we establish arithmetical finiteness results, in particular, for the monotone catenary degree and for the structure of sets of lengths and of their unions.

scholarly journals INTEGRAL DOMAINS WHOSE SIMPLE OVERRINGS ARE INTERSECTIONS OF LOCALIZATIONS

2005 ◽  
Vol 04 (02) ◽  
pp. 195-209 ◽  
Author(s):  
MARCO FONTANA ◽  
EVAN HOUSTON ◽  
THOMAS LUCAS
Keyword(s):  
Dedekind Domain ◽  
Quotient Field ◽  
Integral Domains ◽  
Finiteness Conditions ◽  
Integrally Closed ◽  
Prufer Domains ◽  
Prüfer Domains

Call a domain R an sQQR-domain if each simple overring of R, i.e., each ring of the form R[u] with u in the quotient field of R, is an intersection of localizations of R. We characterize Prüfer domains as integrally closed sQQR-domains. In the presence of certain finiteness conditions, we show that the sQQR-property is very strong; for instance, a Mori sQQR-domain must be a Dedekind domain. We also show how to construct sQQR-domains which have (non-simple) overrings which are not intersections of localizations.

scholarly journals Abstract commutative ideal theory

1962 ◽  
Vol 12 (2) ◽  
pp. 481-498 ◽  
Author(s):  
R. P. Dilworth
Keyword(s):  
Ideal Theory ◽  
Commutative Ideal

Symmetric ring property on nil ideals

2018 ◽  
Vol 17 (01) ◽  
pp. 1850013
Author(s):  
Tai Keun Kwak ◽  
Yang Lee ◽  
Zhelin Piao ◽  
Young Joo Seo
Keyword(s):  
Ideal Theory ◽  
Ring Theory ◽  
Noncommutative Rings ◽  
Commutative Ideal ◽  
Ring Extensions ◽  
Lattice Of Ideals

The usual commutative ideal theory was extended to ideals in noncommutative rings by Lambek, introducing the concept of symmetric. Camillo et al. naturally extended the study of symmetric ring property to the lattice of ideals, defining the new concept of an ideal-symmetric ring. This paper focuses on the symmetric ring property on nil ideals, as a generalization of an ideal-symmetric ring. A ring [Formula: see text] will be said to be right (respectively, left) nil-ideal-symmetric if [Formula: see text] implies [Formula: see text] (respectively, [Formula: see text]) for nil ideals [Formula: see text] of [Formula: see text]. This concept generalizes both ideal-symmetric rings and weak nil-symmetric rings in which the symmetric ring property has been observed in some restricted situations. The structure of nil-ideal-symmetric rings is studied in relation to the near concepts and ring extensions which have roles in ring theory.

Note on Abstract Commutative Ideal Theory

1967 ◽  
Vol 74 (6) ◽  
pp. 706 ◽  
Author(s):  
P. J. McCarthy
Keyword(s):  
Ideal Theory ◽  
Commutative Ideal

Abstract commutative ideal theory without chain condition

1976 ◽  
Vol 6 (1) ◽  
pp. 131-145 ◽  
Author(s):  
D. D. Anderson
Keyword(s):  
Chain Condition ◽  
Ideal Theory ◽  
Commutative Ideal
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