×

A note on \(\oplus\)-cofinitely supplemented modules. (English) Zbl 1136.16007

Summary: Let \(R\) be a ring and \(M\) a right \(R\)-module. In this note, we show that a quotient of an \(\oplus\)-cofinitely supplemented module is not in general \(\oplus\)-cofinitely supplemented and prove that if a module \(M\) is an \(\oplus\)-cofinitely supplemented multiplication module with \(\text{Rad}(M)\ll M\), then \(M\) can be written as an irredundant sum of local direct summands of \(M\). As extension of the result of H. Çalışıcı and A. Pancar [Sib. Mat. Zh. 46, No. 2, 460-465 (2005); translation in Sib. Math. J. 46, No. 2, 359-363 (2005; Zbl 1103.16001)], here it is shown that an arbitrary module is cofinitely semiperfect if and only if it is (amply) cofinitely supplemented by supplements which have projective covers.

MSC:

16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16D40 Free, projective, and flat modules and ideals in associative algebras
16L30 Noncommutative local and semilocal rings, perfect rings

Citations:

Zbl 1103.16001
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] H. Calisici and A. Pancar, “Cofinitely semiperfect modules,” Siberian Mathematical Journal, vol. 46, no. 2, pp. 359-363, 2005. · Zbl 1103.16001
[2] H. Calisici and A. Pancar, “\oplus -cofinitely supplemented modules,” Czechoslovak Mathematical Journal, vol. 54(129), no. 4, pp. 1083-1088, 2004. · Zbl 1080.16002 · doi:10.1007/s10587-004-6453-1
[3] R. B. Warfield Jr., “Decomposability of finitely presented modules,” Proceedings of the American Mathematical Society, vol. 25, no. 1, pp. 167-172, 1970. · Zbl 0204.05902 · doi:10.2307/2036549
[4] A. Idelhadj and R. Tribak, “A dual notion of CS-modules generalization,” in Algebra and Number Theory, vol. 208 of Lecture Notes in Pure and Applied Mathematics, pp. 149-155, Marcel Dekker, New York, NY, USA, 2000. · Zbl 0970.16007
[5] A. Idelhadj and R. Tribak, “On some properties of \oplus -supplemented modules,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 69, pp. 4373-4387, 2003. · Zbl 1066.16001 · doi:10.1155/S016117120320346X
[6] Z. A. El-Bast and P. F. Smith, “Multiplication modules,” Communications in Algebra, vol. 16, no. 4, pp. 755-779, 1988. · Zbl 0642.13002 · doi:10.1080/00927878808823601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.