Geodesic
A CLASS OF EXCHANGE RINGS
Glasgow mathematical journal, Tome 50 (2008) no. 3, pp. 509-522

Voir la notice de l'article provenant de la source Cambridge University Press

It is well known that a ring R is an exchange ring iff, for any a ∈ R, a−e ∈ (a2−a)R for some e2 = e ∈ R iff, for any a ∈ R, a−e ∈ R(a2−a) for some e2 = e ∈ R. The paper is devoted to a study of the rings R satisfying the condition that for each a ∈ R, a−e ∈ (a2−a)R for a unique e2 = e ∈ R. This condition is not left–right symmetric. The uniquely clean rings discussed in (W. K. Nicholson and Y. Zhou, Rings in which elements are uniquely the sum of an idempotent and a unit, Glasgow Math. J. 46 (2004), 227–236) satisfy this condition. These rings are characterized as the semi-boolean rings with a restricted commutativity for idempotents, where a ring R is semi-boolean iff R/J(R) is boolean and idempotents lift modulo J(R) (or equivalently, R is an exchange ring for which any non-zero idempotent is not the sum of two units). Various basic properties of these rings are developed, and a number of illustrative examples are given.
DOI : 10.1017/S0017089508004370
Mots-clés : Primary 16U99 
LEE, TSIU-KWEN; ZHOU, YIQIANG. A CLASS OF EXCHANGE RINGS. Glasgow mathematical journal, Tome 50 (2008) no. 3, pp. 509-522. doi: 10.1017/S0017089508004370
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[1] 1.Anderson, D. D. and Camillo, V. P., Commutative rings whose elements are a sum of a unit and idempotent, Comm. Algebra 30 (7) (2002), 3327–3336. Google Scholar | DOI

[2] 2.Ara, P., Extensions of exchange rings, J. Algebra 197 (2) (1997), 409–423. Google Scholar | DOI

[3] 3.Ara, P., Stability properties of exchange rings, International Symposium on Ring Theory, (Kyongju, 1999), 23–42, Trends Math., Birkhaüser Boston, Boston, MA, 2001. Google Scholar | DOI

[4] 4.Ara, P., Goodearl, K. R., O'Meara, K. C. and Pardo, E., Separative cancellation for projective modules over exchange rings, Israel J. Math. 105 (1998), 105–137. Google Scholar | DOI

[5] 5.Camillo, V. P. and Yu, H.-P., Exchange rings, units and idempotents, Comm. Algebra 22 (1994), 4737–4749. Google Scholar | DOI

[6] 6.Chen, J., Nicholson, W. K. and Zhou, Y., Group rings in which every element is uniquely the sum of a unit and an idempotent, J. Algebra 306 (2) (2006), 453–460. Google Scholar | DOI

[7] 7.Crawley, P. and Jonsson, B., Refinements for infinite decompositions of algebraic systems, Pacific J. Math. 14 (1964), 797–855. Google Scholar | DOI

[8] 8.Dubrovin, N., Příhoda, P., and Puninski, G., Projective modules over the Gerasimov–Sakhaev counter example, J. Algebra 319 (8) (2008), 3259–3279. Google Scholar | DOI

[9] 9.Harada, M., Factor categories with applications to direct sum decomposition of modules, Lecture Notes in Pure and Applied Math. , Marcel Dekker, New York, 1983. Google Scholar

[10] 10.Lam, T. Y., A crash course on stable range, cancellation, substitution and exchange, J. Algebra Appl. 3 (2004), 301–343. Google Scholar | DOI

[11] 11.Levitzki, J., On the structure of algebraic algebras and related rings, Trans. AMS. 74 (1953), 384–409. Google Scholar | DOI

[12] 12.Mohamed, S. H. and Müller, B. J., Continuous and discrete modules, London Math. Soc. Lectures Note Series , Cambridge University Press, Cambridge, UK, 1990. Google Scholar | DOI

[13] 13.Nicholson, W. K., Lifting idempotents and exchange rings, Trans. AMS. 229 (1977), 269–278. Google Scholar | DOI

[14] 14.Nicholson, W. K., On exchange rings, Comm. Algebra 25 (6) (1997), 1917–1918. Google Scholar | DOI

[15] 15.Nicholson, W. K., Strongly clean rings and Fitting's lemma, Comm. Algebra 27 (1999), 3583–3592. Google Scholar | DOI

[16] 16.Nicholson, W. K. and Zhou, Y., Rings in which elements are uniquely the sum of an idempotent and a unit, Glasgow Math. J. 46 (2004), 227–236. Google Scholar | DOI

[17] 17.Nicholson, W. K. and Zhou, Y., Strong lifting, J. Algebra 285 (2) (2005), 795–818. Google Scholar | DOI

[18] 18.Nicholson, W. K. and Zhou, Y., Clean general rings, J. Algebra 291 (1) (2005), 297–311. Google Scholar | DOI

[19] 19.Oshiro, K. and Rizvi, S. T., The exchange property of quasi-continuous modules with the finite exchange property, Osaka J. Math. 33 (1996), 217–234. Google Scholar

[20] 20.Pedersen, G. K. and Perera, P., Inverse limits of rings and multiplier rings, Math. Proc. Camb. Phil. Soc. 139 (2005), 207–228. Google Scholar | DOI

[21] 21.Sánchez, E. Campos, On strongly clean rings, 2002 (unpublished). Google Scholar

[22] 22.Warfield, R. B. Jr., Exchange rings and decompositions of modules, Math. Ann. 199 (1972), 31–36. Google Scholar | DOI

[23] 23.Zimmermann-Huisgen, B. and Zimmermann, W., Classes of modules with the exchange property, J. Algebra 88 (1984), 416–434. Google Scholar | DOI

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