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On a recent generalization of semiperfect rings. (English) Zbl 1159.16015

A classical characterization of semiperfect rings, proved by F. Kasch and E. A. Mares [in Nagoya Math. J. 27, 525-529 (1966; Zbl 0158.28901)] states that a ring is semiperfect if and only if every finitely generated left module \(M\) is supplemented, i.e., for every submodule \(N\leq M\) there is an \(L\leq M\) such that \(L+N=M\), and \(L\) is minimal with this property. So generalizations of the notion of supplemented module lead to generalizations of semiperfect rings. One of these generalizations is the following: a submodule \(N\leq M\) is ‘generalized supplemented’ or ‘Rad-supplemented’ if there is \(L\leq M\) such that \(M=N+L\) and \(N\cap L\subseteq\text{Rad}(M)\).
Recently Y. Wang and N. Ding stated [in Taiwanese J. Math. 10, No. 6, 1589-1601 (2006; Zbl 1122.16003)] that any ring which is generalized supplemented as a left module over itself is semiperfect. The main aim of the present paper is to show that the statement of Wang and Ding is false (Section 2). Moreover, the mentioned statement is corrected: A finitely generated module is generalized supplemented if and only if it is a sum of finitely many modules which have unique maximal submodules (Corollary 3.8). The paper also contains many interesting and illuminating examples.

MSC:

16L30 Noncommutative local and semilocal rings, perfect rings
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16D10 General module theory in associative algebras
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References:

[1] Kasch, Nagoya Math. J. 27 pp 525– (1966) · Zbl 0158.28901 · doi:10.1017/S0027763000026350
[2] Gerasimov, Sibirsk. Mat. Zh. 25 pp 31– (1984) · Zbl 0556.35117 · doi:10.1007/BF00969506
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[7] DOI: 10.1016/j.jpaa.2003.10.012 · Zbl 1055.16001 · doi:10.1016/j.jpaa.2003.10.012
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[9] DOI: 10.1081/AGB-100002396 · Zbl 0989.16001 · doi:10.1081/AGB-100002396
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[11] DOI: 10.2307/1995475 · Zbl 0215.09101 · doi:10.2307/1995475
[12] Wang, Taiwanese J. Math 10 pp 1589– (2006)
[13] DOI: 10.2307/1993568 · Zbl 0094.02201 · doi:10.2307/1993568
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