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Factorization in commutative rings with zero divisors. (English) Zbl 0865.13001
The aim of the paper is to generalize to commutative rings with zero divisors, \(R\), some known facts about factorization into irreducible elements and uniqueness properties which arise for commutative integral domains. Moreover, the paper unifies some of these results which have already been done. In section 2 of the paper under review, several concepts of irreducibility are given in such a way that each one of them have more requirements than the former. These definitions provide their corresponding forms of atomicity which are considered in section 3. Section 4 is devoted to introduce and study the concept of \((\alpha,\beta)\)-unique factorization ring where \(\alpha\) ranges over the different forms of atomicity and \(\beta\) over the forms of factorization as products of the different forms of irreducibility. Finally, the authors give several examples in section 5 and study the extension of the irreducibility and factorization to the polynomial ring \(R[X]\) and the power series ring \(R[[X]]\).

MSC:
13A05 Divisibility and factorizations in commutative rings
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
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[1] J.-C. Allard, Demi-groupes \(D\)-atomiques , C.R. Acad. Sci. Paris 273 (1971), 661-664. · Zbl 0242.20068
[2] D.D. Anderson, The Krull intersection theorem , Pacific J. Math. 57 (1975), 11-14. · Zbl 0322.13004 · doi:10.2140/pjm.1975.57.11
[3] ——–, Some finiteness conditions on a commutative ring , Houston J. Math. 4 (1978), 289-299. · Zbl 0401.13009
[4] ——–, Globalization of some local properties in Krull domains , Proc. Amer. Math. Soc. 85 (1982), 141-145. JSTOR: · Zbl 0498.13009 · doi:10.2307/2044267 · links.jstor.org
[5] D.D. Anderson and D.F. Anderson, Elasticity of factorizations in integral domains , J. Pure Appl. Algebra 80 (1992), 217-235. · Zbl 0773.13003 · doi:10.1016/0022-4049(92)90144-5
[6] D.D. Anderson, D.F. Anderson and R. Markanda, The rings \(R(X)\) and \(R\langle X\rangle\) , J. Algebra 95 (1985), 96-115. · Zbl 0621.13008 · doi:10.1016/0021-8693(85)90096-1
[7] D.D. Anderson, D.F. Anderson and M. Zafrullah, Factorization in integral domains , J. Pure Appl. Algebra 69 (1990), 1-19. · Zbl 0727.13007 · doi:10.1016/0022-4049(90)90074-R
[8] ——–, Factorization in integral domains , II, J. Algebra 152 (1992), 78-93. · Zbl 0776.13001 · doi:10.1016/0021-8693(92)90089-5
[9] ——–, Atomic domains in which almost all atoms are prime , Comm. Alg. 20 (1992), 1447-1462. · Zbl 0749.13012 · doi:10.1080/00927879208824413
[10] ——–, Unique factorization rings with zero divisors : Corrigendum , Houston J. Math. 11 (1985), 423-426. · Zbl 0602.13007
[11] D.D. Anderson and J. Mott, Cohen-Kaplansky domains : integral domains with a finite number of irreducible elements , J. Algebra 148 (1992), 17-41. · Zbl 0773.13005 · doi:10.1016/0021-8693(92)90234-D
[12] M. Billis, Unique factorization in the integers modulo \(n\) , Amer. Math. Monthly 75 (1968), 527. · Zbl 0167.03904 · doi:10.2307/2314728
[13] A. Bouvier, Demi-groupes de type \((R)\). Demi-groupes commutatifs à factorisation unique , C.R. Acad. Sci. Paris 268 (1969), 372-375. · Zbl 0169.03001
[14] ——–, Demi-groupes de type \((R)\) , C.R. Acad. Sci. Paris 270 (1969), 561-563. · Zbl 0198.34201
[15] ——–, Factorisation dans les demi-groupes , C.R. Acad. Sci. Paris 271 (1970), 533-535. · Zbl 0201.02701
[16] ——–, Factorisation dans les demi-groupes de fractions , C.R. Acad. Sci. Paris 271 (1970), 924-925. · Zbl 0208.03001
[17] ——–, Anneaux présimplifiables et anneaux atomiques , C.R. Acad. Sci. Paris 272 (1971), 992-994. · Zbl 0216.32804
[18] ——–, Anneaux de Gauss , C.R. Acad. Sci. Paris 273 (1971), 443-445. · Zbl 0219.13019
[19] ——–, Sur les anneaux de fractions des anneaux atomiques présimplifiables , Bull. Sci. Math. 95 (1971), 371-377. · Zbl 0219.13020
[20] ——–, Remarques sur la facatorisation dans les anneaux commutatifs , Pub. Dépt. Math. Lyon 8 (1971), 1-18. · Zbl 0268.13017
[21] ——–, Anneaux présimplifiable , C.R. Acad. Sci. Paris 274 (1972), 1605-1607. · Zbl 0244.13009
[22] ——–, Résultats nouveaux sur les anneaux présimplifiables , C.R. Acad. Sci. Paris 275 (1972), 955-957. · Zbl 0242.13002
[23] ——–, Anneaux preśimplifiables , Rev. Roum. Math. Pure et Appl. 29 (1974), 713-724. · Zbl 0289.13010
[24] ——–, Structure des anneaux à factorisation unique , Pub. Dépt. Math. Lyon 11 (1974), 39-49. · Zbl 0294.13013
[25] ——–, Sur les anneaux principaux , Acta. Math. Sci. Hung. 27 (1976), 231-242. · Zbl 0336.13010 · doi:10.1007/BF01902099
[26] V. Camillo and R. Guralnick, Polynomial rings over Goldie rings are often Goldie , Proc. Amer. Math. Soc. 98 (1986), 567-568. JSTOR: · Zbl 0608.16020 · doi:10.2307/2045726 · links.jstor.org
[27] C.R. Fletcher, Unique factorization rings , Proc. Camb. Phil. Soc. 65 (1969), 579-583. · Zbl 0174.33401 · doi:10.1017/S0305004100003352
[28] ——–, The structure of unique factorization rings , Proc. Camb. Phil. Soc. 67 (1970), 535-540. · Zbl 0192.38401 · doi:10.1017/S0305004100045825
[29] ——–, Euclidean rings , J. London Math. Soc. 41 (1971), 79-82. · Zbl 0218.13025 · doi:10.1112/jlms/s2-4.1.79
[30] S. Galovich, Unique factorization rings with zero divisors , Math. Magazine 51 (1978), 276-283. JSTOR: · Zbl 0407.13013 · doi:10.2307/2690246 · links.jstor.org
[31] R. Gilmer, Multiplicative ideal theory , Marcel Dekker, New York, 1972. · Zbl 0248.13001
[32] A. Grams, Atomic domains and the ascending chain condition for principal ideals , Proc. Camb. Phil. Soc. 75 (1974), 321-329. · Zbl 0287.13002 · doi:10.1017/S0305004100048532
[33] W. Heinzer and D. Lantz, Commutative rings and ACC on \(n\)-generated ideals, J. Algebra 80 (1983), 261-278. · Zbl 0512.13011 · doi:10.1016/0021-8693(83)90031-5
[34] J. Huckaba, Commutative rings with zero divisors , Marcel Dekker, New York, 1988. · Zbl 0637.13001
[35] I. Kaplansky, Commutative rings , revised edition, University of Chicago, Chicago, 1974. · Zbl 0296.13001
[36] R.E. Kennedy, Krull rings , Pacific J. Math. 89 (1980), 131-136. · Zbl 0402.13012 · doi:10.2140/pjm.1980.89.131
[37] J.W. Kerr, Very long chains of annihilator ideals , Israel J. Math. 47 (1983), 197-204. · Zbl 0528.16007 · doi:10.1007/BF02761952
[38] R. Matsuda, On Kennedy’s problem , Comm. Math. Univ. St. Pauli 31 (1982), 143-145. · Zbl 0502.13009
[39] ——–, Generalizations of multiplicative ideal theory to rings with zero divisors , Bull. Fac. Sci. Ibaraki Univ. Ser. A. Math. 17 (1985), 49-101. · Zbl 0573.13003 · doi:10.5036/bfsiu1968.17.49
[40] S. Mori, Über die produktzerlegung der hauptideal I, J. Sci. Hiroshima Univ. 8 (1938), 7-13. · Zbl 0018.20003
[41] ——–, Über die produktzerlegung der hauptideal II, J. Sci. Hiroshima Univ. 9 (1939), 145-155. · Zbl 0024.00901
[42] ——–, Über die produktzerlegung der hauptideal III, J. Sci. Hiroshima Univ. 10 (1940), 85-94.
[43] ——–, Über die produktzerlegung der hauptideal IV, J. Sci. Hiroshima Univ. 11 (1941), 7-14. · Zbl 0025.01401
[44] M. Roitman, Polynomial extensions of atomic domains , J. Pure Appl. Algebra 87 (1993), 187-199 · Zbl 0780.13014 · doi:10.1016/0022-4049(93)90122-A
[45] P. Samuel, About Euclidean rings , J. Algebra 19 (1971), 282-301. · Zbl 0223.13019 · doi:10.1016/0021-8693(71)90110-4
[46] A. Zaks, Atomic rings without a.c.c. on principal ideals , J. Algebra 74 (1982), 223-231. · Zbl 0484.13012 · doi:10.1016/0021-8693(82)90015-1
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