Cancellation properties in ideal systems: A classification of e.a.b. semistar operations. (English) Zbl 1187.13003

Let \(D\) be a domain with quotient field \(K\) and let \(\star\) be a semistar operation on \(D\). Denote by \(f (D)\) (respectively, \(\overline{F} (D)\)) the set of all finitely generated fractional ideals of \(D\) (respectively, nonzero \(D\)-submodule of \(K\)). A finitely generated fractional ideal \(F\) of \(D\) is called \(\star\)-eab (respectively, \(\star\)-ab) if \((F G)^\star\subset (F H)^\star\) implies that \(G^\star\subset H^\star\), with \(G, H\in f (D)\) (respectively, with \(G, H\in\overline{F} (D)\)) and \(\star\) is called an eab (respectively, ab) semistar operation if each \(F\in f (D)\) is \(\star\)-eab. Let \(\mathfrak w\) be a family of valuation overrings of \(D\). For any \(E\in\overline{F} (D)\), the authors consider the following semistar operations on \(D\): \[ E^{w} =\bigcap \{EW \mid W \in w\}. \] Thus \(w\) is a semistar operation on \(D\). Let \(\star\) be a semistar operation on \(R\). An overring \(V\) of \(D\) is called a \(\star\)-valuation overring of \(D\) if \(F^\star\subset F V\) for all \(F\in f (D)\). The authors define also \(E ^{b(\star)} = \bigcap\{EV \}\), where \(V\) ranges over \(\star\)-valuation overrings of \(R\). Then \(b(\star)\) is also a semistar operation, which is called the complete \(w\)-operation associated with \(\star\). In this paper, the authors prove the implications \(\{\star\) is a complete \(w\)-operation\(\}\Rightarrow \{\star\) is a \(w\)-operation\(\}\Rightarrow \{\star\) is an ab operation\(\}\Rightarrow \{\star\) is an eab operation\(\}\) and the authors point out by giving examples that each of the implications is not reversible.


13A15 Ideals and multiplicative ideal theory in commutative rings
13G05 Integral domains
13F30 Valuation rings
13E99 Chain conditions, finiteness conditions in commutative ring theory
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[1] Gilmer, R., Multiplicative ideal theory, (1972), M. Dekker New York · Zbl 0248.13001
[2] Fontana, M.; Loper, A., A generalization of Kronecker function rings and Nagata rings, Forum math., 19, 971-1004, (2007) · Zbl 1142.13020
[3] Krull, W., Beiträge zur arithmetik kommutativer integritätsbereiche I - II, Math. Z., 41, 545-577, 665-679, (1936) · JFM 62.1105.02
[4] Krull, W., ()
[5] Gilmer, R., Multiplicative ideal theory, part I and part II, Queen’s papers in pure appl. math., 12, (1968)
[6] Anderson, D.D.; Anderson, D.F., Some remarks on cancellation ideals, Math. japonica, 6, 879-886, (1984) · Zbl 0552.13001
[7] Gilmer, R., The cancellation law for ideals in commutative rings, Canad. J. math., 17, 281-287, (1965) · Zbl 0142.00704
[8] I. Kaplansky, Topics in commutative ring theory, Unpublished notes, Chicago, 1971
[9] Jaffard, P., LES systèmes d’idéaux, (1960), Dunod Paris · Zbl 0101.27502
[10] Jensen, C.H., On characterizations of Prüfer domains, Math. scand., 13, 90-98, (1963) · Zbl 0131.27703
[11] Anderson, D.D.; Roitman, M., A characterization of cancellation ideals, Proc. amer. math. soc., 125, 2853-2854, (1997) · Zbl 0883.13001
[12] H.P. Goeters, B. Olberding, On the multiplicative properties of submodules of the quotient field of an integral domain 26 (2000) 241-254 · Zbl 1079.13511
[13] Fanggui, Wang; McCasland, R.L., On \(w\)-modules over strong Mori domains, Comm. algebra, 25, 1285-1306, (1997) · Zbl 0895.13010
[14] Fanggui, Wang; McCasland, R.L., On strong Mori domains, J. pure appl. algebra, 135, 155-165, (1999) · Zbl 0943.13017
[15] Zariski, O.; Samuel, P., Commutative algebra, vol. II, (1960), Van Nostrand New York · Zbl 0121.27801
[16] Fontana, M.; Loper, K.A., Kronecker function rings: A general approach, (), 189-205 · Zbl 1042.13002
[17] Fontana, M.; Loper, K.A., A Krull-type theorem for the semistar integral closure of an integral domain commutative algebra, AJSE arab. J. sci. eng. sect. C theme issues, 26, 89-95, (2001) · Zbl 1271.13020
[18] Halter-Koch, F., Generalized integral closures, (), 349-358 · Zbl 1017.13003
[19] Halter-Koch, F., Ideal systems: an introduction to multiplicative ideal theory, (1998), M. Dekker New York · Zbl 0953.13001
[20] Anderson, D.F.; Houston, E.G.; Zafrullah, M., Pseudo-integrality, Canad. math. bull., 34, 15-22, (1991) · Zbl 0687.13005
[21] Fontana, M.; Loper, K.A., Nagata rings Kronecker function rings and related semistar operations, Comm. algebra, 31, 4775-4805, (2003) · Zbl 1065.13012
[22] Fontana, M.; Huckaba, J.A., Localizing systems and semistar operations, (), 169-198 · Zbl 1047.13002
[23] Fontana, M.; Picozza, G., Semistar invertibility on integral domains, Algebra colloq., 12, 645-664, (2005) · Zbl 1120.13006
[24] Fontana, M.; Jara, P.; Santos, E., Prüfer \(\star\)-multiplication domains and semistar operations, J. algebra appl., 2, 21-50, (2003) · Zbl 1055.13002
[25] Gilmer, R., Overrings of Prüfer domains, J. algebra, 4, 331-340, (1966) · Zbl 0146.26205
[26] Loper, A., More almost Dedekind domains and Prüfer domains of polynomials, (), 287-298 · Zbl 0886.13009
[27] Loper, A., Sequence domains and integer-valued polynomials, J. pure appl. algebra, 119, 185-210, (1997) · Zbl 0960.13005
[28] Loper, A., Almost Dedekind domains that are not Dedekind, (), 276-292 · Zbl 1115.13026
[29] Okabe, A.; Matsuda, R., Semistar operations on integral domains, Math. J. toyama univ., 17, 1-21, (1994) · Zbl 0839.13003
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