On quotients of Rees algebras. (English) Zbl 1499.13014
Following [V. Barucci et al., Commun. Algebra 43, No. 1, 130–142 (2015; Zbl 1327.13087); Ark. Mat. 54, No. 2, 321–338 (2016; Zbl 1372.13017)] the authors of the paper, investigate on a family of quadratic quotients of Rees algebras over a commutative ring. Namely, given a ring \(A\), an ideal \(a\) of \(A\) and a monic polynomial \(f = T^2+aT +b \in A[T]\), the quadratic quotient \(A_f (a)\) of \(A[aT]\) with respect to \(f\) is the factor ring of \(A[aT]\) with respect to its ideal \(fA[T] \cap A[aT]\). More precisely, they answer when the ring \(A_f (a)\) is Noetherian, local, integral domain, complete and Cohen-Macaulay.
Reviewer: Majid Eghbali (Tehran)
References:
| [1] | Atiyah, M. F.; MacDonald, I. G., Introduction to Commutative Algebra (1969), Addison-Wesley · Zbl 0175.03601 |
| [2] | Barucci, V.; D’Anna, M.; Strazzanti, F., A family of quotients of the Rees algebra, Commun. Algebra, 43, 1, 130-142 (2015) · Zbl 1327.13087 |
| [3] | Barucci, V.; D’Anna, M.; Strazzanti, F., Families of Gorenstein and almost Gorenstein rings, Ark. Mat., 54, 2, 321-338 (2016) · Zbl 1372.13017 |
| [4] | Bruns, W.; Herzog, J., Cohen-Maculay Rings (1993), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0788.13005 |
| [5] | D’Anna, M., A construction of Gorenstein rings, J. Algebra, 306, 2, 507-519 (2006) · Zbl 1120.13022 |
| [6] | D’Anna, M.; Finocchiaro, C. A.; Fontana, M., New algebraic properties of an amalgamated algebra along an ideal, Commun. Algebra, 44, 1836-1851 (2016) · Zbl 1345.13002 |
| [7] | D’Anna, M.; Fontana, M., An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl., 6, 3, 443-459 (2007) · Zbl 1126.13002 |
| [8] | D’Anna, M.; Strazzanti, F., New algebraic properties of quadratic quotients of the Rees algebra, J. Algebra Appl., 18, 3, Article 1950047 pp. (2019) · Zbl 1451.13029 |
| [9] | Dobbs, D.; Papick, I., Flat or open implies going down, Proc. Am. Math. Soc., 65, 2, 370-371 (1977) · Zbl 0369.13004 |
| [10] | Fontana, M., Topologically defined classes of commutative rings, Ann. Mat. Pura Appl. (4), 123, 331-355 (1980) · Zbl 0443.13001 |
| [11] | Nagata, M., Local Rings (1975), R. E. Krieger Pub. Co. · Zbl 0386.13010 |
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