$\mathcal{L}$-STABLE RINGS
$\mathcal{L}$-STABLE RINGS
If $\mathcal{L}(R)$ is a set of left ideals defined in
any ring $R,$ we say that $R$ is $\mathcal{L}$-stable if it has stable range
1 relative to the set $\mathcal{L}(R)$. We explore $\mathcal{L}$-stability
in general, characterize when it passes to related classes of rings, and
explore which classes of rings are $\mathcal{L}$-stable for some$\mathcal{\ L}.$ Some well known examples of $\mathcal{L}$-stable rings are presented,
and we show that the Dedekind finite rings are $\mathcal{L}$-stable for a
suitable $\mathcal{L}$.
Keywords:
Stable range, uniquely generated ring, internal cancellation ring, von Neumann regular ring, unit-regular ring, triangular matrix ring left idealtors, $\mathcal{L}$-stable ring,
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- D. D. Anderson, M. Axtell, S. J. Forman and J. Stickles, When are associates unit multiples?, Rocky Mountain J. Math., 34 (2004), 811-828.
- H. Bass, K-Theory and stable algebra, Inst. Hautes tudes Sci. Publ. Math., 22 (1964), 5-60.
- V. Camillo and H.-P. Yu, Exchange rings, units and idempotents, Comm. Algebra, 22 (1994), 4737-4749.
- M. J. Canfell, Completion of diagrams by automorphisms and Bass' first stable range condition, J. Algebra, 176 (1995), 480-503.
- H. Chen, On partially unit-regularity, Kyungpook Math. J., 42 (2002), 13-19.
- H. Chen and W. K. Nicholson, Stable modules and a theorem of Camillo and Yu, J. Pure Appl. Algebra, 218 (2014), 1431-1442.
- G. Ehrlich, Units and one-sided units in regular rings, Trans. Amer. Math. Soc., 216 (1976), 81-90.
- D. Estes and J. Ohm, Stable range in commutative rings, J. Algebra, 7 (1967), 343-362.
- K. R. Goodearl, Von Neumann Regular Rings, Second Edition, Krieger Publishing Co., Malabar, 1991.
- H. K. Grover and D. Khurana, Some characterizations of VNL rings, Comm. Algebra, 37 (2009), 3288-3305.
- I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc., 66 (1949), 464-491.
- D. Khurana and T. Y. Lam, Rings with internal cancellation, J. Algebra, 284 (2005), 203-235.
- T.Y. Lam, A crash course on stable range, cancellation, substitution and exchange, J. Algebra Appl., 3(3) (2004), 301-343.
- W. K. Nicholson, Rings whose elements are quasi-regular or regular, Aequationes Math., 9 (1973), 64-70.
- W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229 (1977), 269-278.
- W. K. Nicholson, On exchange rings, Comm. Algebra, 25 (1997), 1917-1918.
- W. K. Nicholson, Annihilator-stability and unique generation, J. Pure Appl. Algebra, 221 (2017), 2557-2572.
- W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge University Press, Cambridge, 2003.
- L. N. Vaserstein, Bass's first stable range condition, J. Pure Appl. Algebra, 34 (1984), 319-330.
- R. B. Warfield, Exchange rings and decompositions of modules, Math. Ann., 199 (1972), 31-36.
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI