×

On transfer of annihilator conditions of rings. (English) Zbl 1409.16035

Summary: It is well known that a polynomial \(f(x)\) over a commutative ring \(R\) with identity is a zero-divisor in \(R [x]\) if and only if \(f(x)\) has a non-zero annihilator in the base ring, where \(R [x]\) is the polynomial ring with indeterminate \(x\) over \(R\). But this result fails in non-commutative rings and in the case of formal power series ring. In this paper, we consider the problem of determining some annihilator properties of the formal power series ring \(R [[x]]\) over an associative non-commutative ring \(R\). We investigate relations between power series-wise McCoy property and other standard ring-theoretic properties. In this context, we consider right zip rings, right strongly \(A B\) rings and rings with right Property \((A)\). We give a generalization (in the case of non-commutative ring) of a classical results related to the annihilator of formal power series rings over the commutative Noetherian rings. We also give a partial answer, in the case of formal power series ring, to the question posed in [the first author and D. Kiani, J. Algebra Appl. 13, No. 2, Article ID 1350083, 23 p. (2014; Zbl 1295.16026), Question, p. 16].

MSC:

16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16P40 Noetherian rings and modules (associative rings and algebras)

Citations:

Zbl 1295.16026
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alhevaz, A.; Kiani, D., Mccoy property of skew Laurent polynomials and power series rings, J. Algebra Appl., 13, 2, 23, (2014) · Zbl 1295.16026
[2] Bell, H. E., Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc., 2, 363-368, (1970) · Zbl 0191.02902
[3] Camillo, V.; Nielsen, P. P., Mccoy rings and zero-divisors, J. Pure Appl. Algebra, 212, 3, 599-615, (2008) · Zbl 1162.16021
[4] Cedó, F., Zip rings and mal’cev domains, Comm. Algebra, 19, 7, 1983-1991, (1991) · Zbl 0733.16007
[5] Cohn, P. M., Reversible rings, Bull. London Math. Soc., 31, 6, 641-648, (1999) · Zbl 1021.16019
[6] Faith, C., Rings with zero intersection property on annihilators: ZIP rings, Publ. Math., 33, 2, 329-338, (1989) · Zbl 0702.16015
[7] Faith, C., Annihilator ideals, associated primes and kasch-mccoy commutative rings, Comm. Algebra, 19, 7, 1867-1892, (1991) · Zbl 0729.16015
[8] Feller, E. H., Properties of primary noncommutative rings, Trans. Amer. Math. Soc., 89, 79-91, (1958) · Zbl 0095.25703
[9] Fields, D. E., Zero divisors and nilpotent elements in power series rings, Proc. Amer. Math. Soc., 27, 427-433, (1971) · Zbl 0219.13023
[10] Gilmer, R.; Grams, A.; Parker, T., Zero divisors in power series rings, J. Reine Angew. Math., 278/279, 145-164, (1975) · Zbl 0309.13009
[11] Habeb, J. M., A note on zero commutative and duo rings, Math. J. Okayama Univ., 32, 73-76, (1990) · Zbl 0758.16007
[12] Hashemi, E., Mccoy rings relative to a monoid, Comm. Algebra, 38, 3, 1075-1083, (2010) · Zbl 1207.16041
[13] Hashemi, E., Extensions of zip rings, Studia Sci. Math. Hung., 47, 4, 522-528, (2010) · Zbl 1221.16019
[14] Hashemi, E.; Estaji, A. As.; Ziembowski, M., Answers to some questions concerning rings with property \((A)\), Proc. Edinb. Math. Soc., 60, 3, 651-664, (2017) · Zbl 1405.16036
[15] Hirano, Y., On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra, 168, 1, 45-52, (2002) · Zbl 1007.16020
[16] Hong, C. Y.; Kim, N. K.; Kwak, T. K.; Lee, Y., Extensions of zip rings, J. Pure Appl. Algebra, 195, 3, 231-242, (2005) · Zbl 1071.16020
[17] Hong, C. Y.; Kim, N. K.; Lee, Y., Extensions of mccoy’s theorem, Glasg. Math. J., 52, 1, 155-159, (2010) · Zbl 1195.16026
[18] Hwang, S. U.; Kim, N. K.; Lee, Y., On rings whose right annihilators are bounded, Glasg. Math. J., 51, 3, 539-559, (2009) · Zbl 1198.16001
[19] Hong, C. Y.; Kim, N. K.; Lee, Y.; Ryu, S. J., Rings with property \((A)\) and their extensions, J. Algebra, 315, 2, 612-628, (2007) · Zbl 1156.16001
[20] Kwak, T. K.; Lee, Y., Zero divisors in skew power series rings, Comm. Algebra, 43, 10, 4427-4445, (2015) · Zbl 1327.16035
[21] Lam, T. Y., A First Course in Noncommutative Rings, 131, (2001), Springer-Verlag, New York · Zbl 0980.16001
[22] Leroy, A.; Matczuk, J., Zip property of certain ring extensions, J. Pure Appl. Algebra, 220, 1, 335-345, (2016) · Zbl 1334.16019
[23] Mason, G., Reflexive ideals, Comm. Algebra, 9, 17, 1709-1724, (1981) · Zbl 0468.16024
[24] McCoy, N. H., Remarks on divisors of zero, Amer. Math. Monthly, 49, 286-295, (1942) · Zbl 0060.07703
[25] McCoy, N. H., Annihilators in polynomial rings, Amer. Math. Monthly, 64, 28-29, (1957) · Zbl 0077.25903
[26] Motais de Narbonne, L., Anneaux semi-commutatifs et unis riels anneaux dont LES id aux principaux sont idempotents, Proc. 106’th Nat. Congress of Learned Societies (Perpignan, 1981), 7173, (1982), Bib. Nat., Paris
[27] Nielsen, P. P., Semicommutativity and the mccoy condition, J. Algebra, 298, 1, 134-141, (2006) · Zbl 1110.16036
[28] Shin, G., Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc., 184, 43-60, (1973) · Zbl 0283.16021
[29] Tuganbaev, A. A., Semidistributive Modules and Rings, 449, (1998), Kluwer Academic Publishers · Zbl 0909.16001
[30] Yang, S. Z.; Song, X. M.; Liu, Z. K., Power-serieswise mccoy rings, Algebra Colloq., 18, 2, 301-310, (2011) · Zbl 1220.16031
[31] Zelmanowitz, J. M., The finite intersection property on annihilator right ideals, Proc. Amer. Math. Soc., 57, 2, 213-216, (1976) · Zbl 0333.16014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.