×

Rings over which cyclics are direct sums of projective and CS or Noetherian. (English) Zbl 1208.16006

A ring \(R\) is called a right \(WV\)-ring if every simple right \(R\)-module is injective relative to proper cyclics. If \(R\) is a right \(WV\)-ring, then it is right uniform or a right \(V\)-ring. Among other results, the authors show that a right \(WV\)-ring is right Noetherian if and only if every right cyclic module is a direct sum of a projective module and a CS module or Noetherian module. Also, for a finitely generated module \(M\) with projective socle over a \(V\)-ring \(R\) such that every subfactor of \(M\) is a direct sum of a projective module and a CS module or Noetherian module, it is proved that \(M\) is a direct sum of a semisimple module and a Noetherian module with zero socle.

MSC:

16D50 Injective modules, self-injective associative rings
16D40 Free, projective, and flat modules and ideals in associative algebras
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16P40 Noetherian rings and modules (associative rings and algebras)
16D80 Other classes of modules and ideals in associative algebras
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Lam, Lectures on modules and rings (1999) · Zbl 0911.16001 · doi:10.1007/978-1-4612-0525-8
[2] Levy, Pacific J. Math. 18 pp 149– (1966) · Zbl 0139.26403 · doi:10.2140/pjm.1966.18.149
[3] DOI: 10.1016/0022-4049(95)00154-9 · Zbl 0857.16003 · doi:10.1016/0022-4049(95)00154-9
[4] DOI: 10.1142/S0219498808002631 · Zbl 1187.16018 · doi:10.1142/S0219498808002631
[5] DOI: 10.2307/2040356 · Zbl 0313.16025 · doi:10.2307/2040356
[6] Faith, Rings and things and a fine array of twentieth century associative algebra (1999) · Zbl 0917.16001
[7] Wisbauer, Foundations of module and ring theory (1991)
[8] Faith, Algebra: rings, modules and categories (1973) · Zbl 0266.16001
[9] Shock, Pacific J. Math. 54 pp 227– (1974) · Zbl 0303.13013 · doi:10.2140/pjm.1974.54.227
[10] DOI: 10.1007/BF01189998 · Zbl 0763.16002 · doi:10.1007/BF01189998
[11] Smith, Can. J. Math. 24 pp 93– (1979) · Zbl 0412.16012 · doi:10.4153/CJM-1979-011-2
[12] Dung, Extending modules (1994)
[13] DOI: 10.1081/AGB-120013181 · Zbl 0998.16006 · doi:10.1081/AGB-120013181
[14] DOI: 10.1016/0021-8693(91)90298-M · Zbl 0737.16001 · doi:10.1016/0021-8693(91)90298-M
[15] DOI: 10.1016/0021-8693(73)90088-4 · Zbl 0258.16023 · doi:10.1016/0021-8693(73)90088-4
[16] DOI: 10.2307/1990539 · Zbl 0035.30301 · doi:10.2307/1990539
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.