Unit-regular modules. (English) Zbl 1437.16007
Summary: In [J. Pure Appl. Algebra 218, No. 8, 1431–1442 (2014; Zbl 1320.16008)], the first two authors proved an extension to modules of a theorem of V. P. Camillo and H.-P. Yu [Trans. Am. Math. Soc. 347, No. 8, 3141–3147 (1995; Zbl 0848.16008)] that an exchange ring has stable range 1 if and only if every regular element is unit-regular. Here, we give a Morita context version of a stronger theorem. The definition of regular elements in a module goes back to
J. Zelmanowitz in [Trans. Am. Math. Soc. 163, 341–355 (1972; Zbl 0227.16022)], but the notion of a unit-regular element in a module is new. In this paper, we study unit-regular elements and give several characterizations of them in terms of “stable” elements and “lifting” elements. Along the way, we give natural extensions to the module case of many results about unit-regular rings. The paper concludes with a discussion of when the endomorphism ring of a unit-regular module is a unit-regular ring.
MSC:
| 16E50 | von Neumann regular rings and generalizations (associative algebraic aspects) |
References:
| [1] | 1.H.Bass, K-Theory and stable algebra, Publ. Math. IHES22 (1964), 5-70.10.1007/BF02684689 · Zbl 0248.18025 · doi:10.1007/BF02684689 |
| [2] | 2.G.Azumaya, Some characterizations of regular modules, Publ. Mat.34 (1950), 241-248.10.5565/PUBLMAT_34290_02 · Zbl 0722.16001 · doi:10.5565/PUBLMAT_34290_02 |
| [3] | 3.V. P.Camillo and D.Khurana, A characterization of unit-regular rings, Comm. Algebra29 (2001), 2293-2295.10.1081/AGB-100002185 · Zbl 0992.16011 |
| [4] | 4.V.Camillo and H.-P.Yu, Stable range 1 for rings with many idempotents, Trans. Amer. Math. Soc.347(8) (1995), 3141-3147.10.1090/S0002-9947-1995-1277100-2 · Zbl 0848.16008 |
| [5] | 5.H.Chen and W. K.Nicholson, Stable modules and a theorem of Camillo and Yu, J. Pure Appl. Algebra218 (2014), 1431-1442.10.1016/j.jpaa.2013.11.026 · Zbl 1320.16008 |
| [6] | 6.G.Ehrlich, Units and one-sided units in regular rings, Trans. Amer. Math. Soc.216 (1976), 81-90.10.1090/S0002-9947-1976-0387340-0 · Zbl 0298.16012 · doi:10.1090/S0002-9947-1976-0387340-0 |
| [7] | 7.F.Siddique, On two questions of Nicholson, preprint. |
| [8] | 8.D.Khurana and T. Y.Lam, Rings with internal cancellation, J. Algebra284 (2005), 203-235.10.1016/j.jalgebra.2004.07.032 · Zbl 1076.16004 |
| [9] | 9.W. K.Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc.229 (1977), 269-278.10.1090/S0002-9947-1977-0439876-2 · Zbl 0352.16006 · doi:10.1090/S0002-9947-1977-0439876-2 |
| [10] | 10.W. K.Nicholson and E.Sánchez Campos, Morphic modules, Comm. Algebra33(8) (2005), 2629-2647.10.1081/AGB-200064348 · Zbl 1084.16005 |
| [11] | 11.L. N.Vaserstein, Bass’s first stable range condition, J. Pure. Appl. Algebra34 (1984), 319-330.10.1016/0022-4049(84)90044-6 · Zbl 0547.16017 · doi:10.1016/0022-4049(84)90044-6 |
| [12] | 12.R. B.Warfield, Exchange rings and decompositions of modules, Math. Ann.199 (1972), 460-465.10.1007/BF01419573 · Zbl 0228.16012 · doi:10.1007/BF01419573 |
| [13] | 13.J. M.Zelmanowitz, Regular modules, Trans. Amer. Math. Soc.163 (1972), 341-355.10.1090/S0002-9947-1972-0286843-3 · Zbl 0227.16022 · doi:10.1090/S0002-9947-1972-0286843-3 |
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