# Article

Full entry | PDF   (0.2 MB)
Keywords:
almost principal ideal; divisorial ideal; greatest common divisor domain; Schreier domain; uppers to zero
Summary:
Let $R$ be an integral domain with quotient field $K$ and $f(x)$ a polynomial of positive degree in $K[x]$. In this paper we develop a method for studying almost principal uppers to zero ideals. More precisely, we prove that uppers to zero divisorial ideals of the form $I = f(x)K[x] \cap R[x]$ are almost principal in the following two cases: – $J$, the ideal generated by the leading coefficients of $I$, satisfies $J^{-1} = R$. – $I^{-1}$ as the $R[x]$-submodule of $K(x)$ is of finite type. Furthermore we prove that for $I = f(x)K[x] \cap R[x]$ we have: – $I^{-1}\cap K[x]=(I:_{K(x)}I)$. – If there exists $p/q \in I^{-1}-K[x]$, then $(q,f)\neq 1$ in $K[x]$. If in addition $q$ is irreducible and $I$ is almost principal, then $I' = q(x)K[x] \cap R[x]$ is an almost principal upper to zero. Finally we show that a Schreier domain $R$ is a greatest common divisor domain if and only if every upper to zero in $R[x]$ contains a primitive polynomial.
References:
[1] Anderson, D. D., Anderson, D. F.: Generalized GCD domains. Comment. Math. Univ. St. Pauli 28 (1980), 215-221. MR 0578675 | Zbl 0434.13001
[2] Anderson, D. D., Dumitrescu, T., Zafrullah, M.: Quasi-Schreier domains. II. Commun. Algebra 35 (2007), 2096-2104. DOI 10.1080/00927870701302107 | MR 2331832 | Zbl 1119.13001
[3] Anderson, D. D., Zafrullah, M.: The Schreier property and Gauss' Lemma. Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 10 (2007), 43-62. MR 2310957 | Zbl 1129.13025
[4] Cohn, P. M.: Bezout rings and their subrings. Proc. Camb. Philos. Soc. 64 (1968), 251-264. DOI 10.1017/S0305004100042791 | MR 0222065 | Zbl 0157.08401
[5] Gilmer, R.: Multiplicative Ideal Theory. Pure and Applied Mathematics. Vol. 12. Marcel Dekker New York (1972). MR 0427289
[6] Hamann, E., Houston, E., Johnson, J. L.: Properties of uppers to zero in $R[x]$. Pac. J. Math. 135 (1988), 65-79. DOI 10.2140/pjm.1988.135.65 | MR 0965685 | Zbl 0627.13007
[7] Houston, E.: Uppers to zero in polynomial rings. Multiplicative Ideal Theory in Commutative Algebra. A Tribute to the Work of Robert Gilmer J. W. Brewer et al. Springer New York (2006), 243-261. MR 2265813 | Zbl 1116.13014
[8] Houston, E., Zafrullah, M.: UMV-domains. Arithmetical Properties of Commutative Rings and Monoids. Lecture Notes in Pure and Applied Mathematics 241 S. T. Chapman Chapman & Hall/CRC Boca Raton (2005), 304-315. MR 2140703 | Zbl 1079.13015
[9] Kaplansky, I.: Commutative Rings. Allyn and Bacon Boston (1970). MR 0254021 | Zbl 0203.34601
[10] Tang, H. T.: Gauss' lemma. Proc. Am. Math. Soc. 35 (1972), 372-376. MR 0302638 | Zbl 0266.13007
[11] Zafrullah, M.: The $D+XD_S[X]$ construction from GCD-domains. J. Pure Appl. Algebra 50 (1988), 93-107. DOI 10.1016/0022-4049(88)90006-0 | MR 0931909
[12] Zafrullah, M.: What $v$-coprimality can do for you. Multiplicative Ideal Theory in Commutative Algebra. A Tribute to the Work of Robert Gilmer J. W. Brewer et al. Springer New York (2006), 387-404. MR 2265821 | Zbl 1138.13001

Partner of