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\noindent Stable range of rings is a unifying concept for problems related to the substitution and cancellation of modules. The newly appeared element-wise setting for the simplest case of stable range one is tempting to study the lifting property modulo ideals. We study the lifting of elements having (idempotent) stable range one from a quotient of a ring $R$ modulo a two-sided ideal $I$ by providing several examples and investigating the relations with other lifting properties, including lifting idempotents, lifting units, and lifting of von Neumann regular elements. In the case where the ring $R$ is a left or a right duo ring, we show that stable range one elements lift modulo every two-sided ideal if and only if $R$ is a ring with stable range one. Under a mild assumption, we further prove that the lifting of elements having idempotent stable range one implies the lifting of von Neumann regular elements.
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참고문헌 (30)
참고문헌 신청
F. W. Anderson / 1992 / Graduate Texts in Mathematics, 13 / Springer-Verlag
G. Baccella / 1995 / Semi-Artinian V-rings and semi-Artinian von Neumann regular rings / J. Algebra 173(3):587~612
G. Baccella / 2002 / Exchange property and the natural preorder between simple modules over semi-Artinian rings / J. Algebra 253(1):133~166
H. Bass / 1964 / K-theory and stable algebra / Inst. Hautes Etudes Sci. Publ. Math 22(22):5~60
H. Chen / 1999 / Rings with many idempotents / Int. J. Math. Math. Sci 22(3):547~558
G. Baccella / 1995 / Semi-Artinian V-rings and semi-Artinian von Neumann regular rings / J. Algebra 173(3):587~612
G. Baccella / 2002 / Exchange property and the natural preorder between simple modules over semi-Artinian rings / J. Algebra 253(1):133~166
H. Bass / 1964 / K-theory and stable algebra / Inst. Hautes Etudes Sci. Publ. Math 22(22):5~60
H. Chen / 1999 / Rings with many idempotents / Int. J. Math. Math. Sci 22(3):547~558
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