The Schreier Property and Gauss' Lemma

Daniel D. Anderson; Muhammad Zafrullah

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 1, page 43-62
  • ISSN: 0392-4041

Abstract

top
Let D be an integral domain with quotient field D . Recall that D is Schreier if D is integrally closed and for all x , y , z ∈ D ∖ { 0 } , x | y z implies that x = r ⋅ s where r | y e s | z . A GCD domain is Schreier. We show that an integral domain D is a GCD domain if and only if (i) for each pair a , b ∈ D ∖ { 0 } , there is a finitely generated ideal B such that a ⁢ D ⁢ ⋂ b ⁢ D = B v and (ii) every quadratic in D ⁢ [ X ] that is a product of two linear polynomials in K ⁢ [ X ] is a product of two linear polynomials in D ⁢ [ X ] . We also show that D is Schreier if and only if every polynomial in D ⁢ [ X ] with a linear factor in K ⁢ [ X ] has a linear factor in D ⁢ [ X ] and show that D is a Schreier domain with algebraically closed quotient field if and only if every nonconstant polynomial over D is expressible as a product of linear polynomials. We also compare the two most common modes of generalizing GCD domains. One is via properties that imply Gauss' Lemma and the other is via the Schreier property. The Schreier property is not implied by any of the specializations of Gauss' Lemma while all but one of the specializations of Gauss Lemma are implied by the Schreier property.

How to cite

top

Anderson, Daniel D., and Zafrullah, Muhammad. "The Schreier Property and Gauss' Lemma." Bollettino dell'Unione Matematica Italiana 10-B.1 (2007): 43-62. <http://eudml.org/doc/290437>.

@article{Anderson2007,
abstract = {Let $D$ be an integral domain with quotient field $D$. Recall that $D$ is Schreier if $D$ is integrally closed and for all $x, y, z \in D \setminus \\{0\\}$, $x|yz$ implies that $x = r \cdot s$ where $r|y$ e $s|z$. A GCD domain is Schreier. We show that an integral domain $D$ is a GCD domain if and only if (i) for each pair $a, b \in D \setminus \\{0\\}$, there is a finitely generated ideal $B$ such that $aD \bigcap bD = B_v$ and (ii) every quadratic in $D[X]$ that is a product of two linear polynomials in $K[X]$ is a product of two linear polynomials in $D[X]$. We also show that $D$ is Schreier if and only if every polynomial in $D[X]$ with a linear factor in $K[X]$ has a linear factor in $D[X]$ and show that $D$ is a Schreier domain with algebraically closed quotient field if and only if every nonconstant polynomial over $D$ is expressible as a product of linear polynomials. We also compare the two most common modes of generalizing GCD domains. One is via properties that imply Gauss' Lemma and the other is via the Schreier property. The Schreier property is not implied by any of the specializations of Gauss' Lemma while all but one of the specializations of Gauss Lemma are implied by the Schreier property.},
author = {Anderson, Daniel D., Zafrullah, Muhammad},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {43-62},
publisher = {Unione Matematica Italiana},
title = {The Schreier Property and Gauss' Lemma},
url = {http://eudml.org/doc/290437},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Anderson, Daniel D.
AU - Zafrullah, Muhammad
TI - The Schreier Property and Gauss' Lemma
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/2//
PB - Unione Matematica Italiana
VL - 10-B
IS - 1
SP - 43
EP - 62
AB - Let $D$ be an integral domain with quotient field $D$. Recall that $D$ is Schreier if $D$ is integrally closed and for all $x, y, z \in D \setminus \{0\}$, $x|yz$ implies that $x = r \cdot s$ where $r|y$ e $s|z$. A GCD domain is Schreier. We show that an integral domain $D$ is a GCD domain if and only if (i) for each pair $a, b \in D \setminus \{0\}$, there is a finitely generated ideal $B$ such that $aD \bigcap bD = B_v$ and (ii) every quadratic in $D[X]$ that is a product of two linear polynomials in $K[X]$ is a product of two linear polynomials in $D[X]$. We also show that $D$ is Schreier if and only if every polynomial in $D[X]$ with a linear factor in $K[X]$ has a linear factor in $D[X]$ and show that $D$ is a Schreier domain with algebraically closed quotient field if and only if every nonconstant polynomial over $D$ is expressible as a product of linear polynomials. We also compare the two most common modes of generalizing GCD domains. One is via properties that imply Gauss' Lemma and the other is via the Schreier property. The Schreier property is not implied by any of the specializations of Gauss' Lemma while all but one of the specializations of Gauss Lemma are implied by the Schreier property.
LA - eng
UR - http://eudml.org/doc/290437
ER -

References

top
  1. ANDERSON, D.D. - ANDERSON, D.F. - ZAFRULLAH, M., Splitting the t-class group, J. Pure Appl. Algebra, 74 (1991), 17-37. Zbl0760.13006MR1129127DOI10.1016/0022-4049(91)90046-5
  2. ANDERSON, D.D. - QUINTERO, R., Some generalizations of GCD-domains, Factorization in Integral Domains, 439-480, Lecture Notes in Pure and Appl. Math., 189, Dekker, New York, 1997. MR1460772
  3. ANDERSON, D.D. - ZAFRULLAH, M., M, P.. Cohn's completely primal elements, Zero-dimensional Commutative Rings, 115-123, Lecture Notes in Pure and Appl. Math., 171, Dekker, New York, 1995. Zbl0887.13012MR1335708
  4. ANDERSON, D.D. - ZAFRULLAH, M., Splitting sets in integral domains, Proc. Amer. Math. Soc., 129 (2001), 2209-2217. Zbl0962.13001MR1823902DOI10.1090/S0002-9939-00-05863-9
  5. ANDERSON, D.F., Integral v-ideals, Glasgow Math. J., 22 (1981), 167-172. Zbl0467.13012MR623001DOI10.1017/S0017089500004638
  6. ARNOLD, J. - SHELDON, P., Integral domains that satisfy Gauss's lemma, Michigan Math. J., 22 (1975), 39-51. Zbl0314.13008MR371887
  7. BASTIDA, E. - GILMER, R., Overrings and divisorial ideals of rings of the form D+M, Michigan Math. J., 20 (1973), 79-95. Zbl0239.13001MR323782
  8. BROOKFIELD, G. - RUSH, D., An antimatter domain that is not pre-Schreier, preprint. 
  9. COHN, P.M., Bezout rings and their subrings, Proc. Cambridge Phil. Soc., 64 (1968), 251-264. Zbl0157.08401MR222065
  10. COYKENDALL, J. - DOBBS, D. - MULLINS, B., On integral domains with no atoms, Comm. Algebra, 27 (1999), 5813-5831. Zbl0990.13015MR1726278DOI10.1080/00927879908826792
  11. DOBBS, D., Coherence, ascent of going-down, and pseudo-valuation domains, Houston J. Math., 4 (1978), 551-567. Zbl0388.13002MR523613
  12. GILMER, R., Multiplicative Ideal Theory, Dekker, New York, 1972. Zbl0248.13001MR427289
  13. GILMER, R. - PARKER, T., Divisibility properties in semigroup rings, Michigan Math. J., 21 (1974), 65-86. Zbl0285.13007MR342635
  14. HEDSTROM, J. - HOUSTON, E., Pseudo-valuation domains, Pacific J. Math., 75 (1978), 137-147. Zbl0368.13002MR485811
  15. KAPLANSKY, I., Commutative Rings, Allyn and Bacon, Boston, 1970. MR254021
  16. MALIK, S. - MOTT, J. - ZAFRULLAH, M., The GCD property and irreducible quadratic polynomials, Internat. J. Math. Sci., 9 (1986), 749-752. Zbl0646.13009MR870530DOI10.1155/S0161171286000893
  17. MOTT, J. - NASHIER, B. - ZAFRULLAH, M., Contents of polynomials and invertibility, Comm. Algebra, 18 (1990), 1569-1583. Zbl0705.13005MR1059749DOI10.1080/00927879008823984
  18. MCADAM, S. - RUSH, D., Schreier rings, Bull. London Math. Soc., 10 (1978), 77-80. MR476723DOI10.1112/blms/10.1.77
  19. MOTT, J. - SCHEXNAYDER, M., Exact sequences of semi-value groups, J. Reine Angew Math.283/284 (1976), 388-401. Zbl0347.13001MR404247
  20. RUSH, D., Quadratic polynomials, factorization in integral domains and Schreier domains in pullbacks, Mathematika, 50 (2003), 103-112. Zbl1076.13002MR2136355DOI10.1112/S0025579300014832
  21. SCHEXNAYDER, M., Groups of Divisibility, Dissertation, Florida State University, 1973. 
  22. WATERHOUSE, W., Quadratic polynomials and unique factorization, Amer. Math. Monthly, 109 (2002), 70-72. Zbl1036.13004MR1903514DOI10.2307/2695769
  23. ZAFRULLAH, M., On a property of pre-Schreier domains, Comm. Algebra, 15 (1987), 1895-1920. Zbl0634.13004MR898300DOI10.1080/00927878708823512
  24. ZAFRULLAH, M., Well-behaved prime t-ideals, J. Pure Appl. Algebra, 65 (1990), 199-207. Zbl0705.13001MR1068255DOI10.1016/0022-4049(90)90119-3
  25. ZAFRULLAH, M., Putting t-invertibility to use, Non-Noetherian Commutative Ring Theory, 429-457, Math. Appl., 520, Kluwer Acad. Publ., Dordrecht, 2000. Zbl0988.13003MR1858174

NotesEmbed ?

top

You must be logged in to post comments.