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Stable modules and a theorem of Camillo and Yu. (English) Zbl 1320.16008

The main goal of the paper is to extend to modules a theorem of Camillo and Yu which states that an exchange ring has the K-theoretic condition stable range \(1\) if and only if all of its (von Neumann) regular elements are unit-regular [V. P. Camillo and H.-P. Yu, Trans. Am. Math. Soc. 347, No. 8, 3141-3147 (1995; Zbl 0848.16008)]. Towards that end, concepts of stability (an analog of stable range \(1\)) and unit-regularity are developed for modules and their elements, and more generally for Morita contexts \(\left[\begin{smallmatrix} R&V\\ W&S\end{smallmatrix}\right]\) and elements of \(V\). Applications to a left \(R\)-module \(M\) are obtained by specializing to the standard context \(\left[\begin{smallmatrix} R&M\\ \operatorname{Hom}(M,R)&\text{End}(M)\end{smallmatrix}\right]\). Regularity for an element \(m\in M\) was defined by J. Zelmanowitz [Trans. Am. Math. Soc. 163, 341-355 (1972; Zbl 0227.16022)] to mean that \(((m)f)m=m\) for some \(f\in\operatorname{Hom}(M,R)\), which is equivalent to saying that the element \(\left[\begin{smallmatrix} 0&m\\ 0&0\end{smallmatrix}\right]\) in the ring of the standard context is regular.
The main theorem states that a module \(M\) with the finite exchange property is stable if and only if every regular element of \(M\) is unit-regular. The authors also develop a form of stability called regular-stability, which is a condition on the regular elements of a module, and they prove that an arbitrary module \(M\) is regular-stable if and only if every regular element of \(M\) is unit-regular. In case \(M={_RR}\), these conditions are equivalent to internal direct sum cancellation, meaning that whenever \(_RR=K_1\oplus L=K_2\oplus L'\) with \(L\cong L'\), it follows that \(K_1\cong K_2\). This equivalence follows from a proof of G. Ehrlich [Trans. Am. Math. Soc. 216, 81-90 (1976; Zbl 0298.16012)], as made clear by D. Khurana and T. Y. Lam [J. Algebra 284, No. 1, 203-235 (2005; Zbl 1076.16004)]. For a general \(R\)-module \(M\), the present authors prove that \(M\) is regular-stable if and only if the following comparability condition holds: whenever \(_RR=K_1\oplus L\) and \(M=K_2\oplus L'\) with \(L\cong L'\), the left ideal \(K_1\) is isomorphic to a direct summand of \(K_2\).

MSC:

16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
19B10 Stable range conditions
16U99 Conditions on elements
16E20 Grothendieck groups, \(K\)-theory, etc.
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16D90 Module categories in associative algebras
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References:

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