\(\delta\)-small submodules and \(\delta\)-supplemented modules. (English) Zbl 1152.16003
Throughout \(R\) is an associative ring with identity and modules are unitary. Let \(M\) be a module and \(N,L\) be submodules of \(M\). Then \(N\) is called: (i) \(\delta\)-small in \(M\) if, whenever \(N+X=M\) with \(M/X\) singular, we have \(X=M\); (ii) \(\delta\)-supplement of \(L\) if \(M=N+L\) and \(N\cap L\) is \(\delta\)-small in \(N\); (iii) \(\delta\)-supplement submodule if \(N\) is a \(\delta\)-supplement for some submodule of \(M\). The module \(M\) is called amply \(\delta\)-supplemented if for any submodules \(A,B\) of \(M\) with \(M=A+B\) there exists a \(\delta\)-supplement \(P\) of \(A\) such that \(P\subseteq B\). Denote by \(\delta(M)\) the reject in \(M\) of the class of all singular simple modules.
The author proves that \(\delta(M)\) is Noetherian (resp. Artinian) if and only if \(M\) satisfies ACC (resp. DCC) on \(\delta\)-small submodules. It is also shown that a module \(M\) is Artinian if and only if \(M\) is amply \(\delta\)-supplemented and satisfies DCC on \(\delta\)-supplement submodules and on \(\delta\)-small submodules.
The author proves that \(\delta(M)\) is Noetherian (resp. Artinian) if and only if \(M\) satisfies ACC (resp. DCC) on \(\delta\)-small submodules. It is also shown that a module \(M\) is Artinian if and only if \(M\) is amply \(\delta\)-supplemented and satisfies DCC on \(\delta\)-supplement submodules and on \(\delta\)-small submodules.
Reviewer: Iuliu Crivei (Cluj-Napoca)
MSC:
| 16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |
| 16P70 | Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras) |
Keywords:
small submodules; amply supplemented modules; DCC on supplemented submodules; singular simple modulesReferences:
| [1] | Y. Zhou, “Generalizations of perfect, semiperfect, and semiregular rings,” Algebra Colloquium, vol. 7, no. 3, pp. 305-318, 2000. · Zbl 0994.16016 · doi:10.1007/s10011-000-0305-9 |
| [2] | E. P. Armendariz, “Rings with DCC on essential left ideals,” Communications in Algebra, vol. 8, no. 3, pp. 299-308, 1980. · Zbl 0444.16015 · doi:10.1080/00927878008822460 |
| [3] | I. Al-Khazzi and P. F. Smith, “Modules with chain conditions on superfluous submodules,” Communications in Algebra, vol. 19, no. 8, pp. 2331-2351, 1991. · Zbl 0734.16009 · doi:10.1080/00927879108824262 |
| [4] | V. Camillo and M. F. Yousif, “CS-modules with ACC or DCC on essential submodules,” Communications in Algebra, vol. 19, no. 2, pp. 655-662, 1991. · Zbl 0718.16006 · doi:10.1080/00927879108824160 |
| [5] | Y. Wang and N. Ding, “Generalized supplemented modules,” Taiwanese Journal of Mathematics, vol. 10, no. 6, pp. 1589-1601, 2006. · Zbl 1122.16003 |
| [6] | R. Wisbauer, Foundations of Module and Ring Theory, vol. 3 of Algebra, Logic and Applications, Gordon and Breach Science, Philadelphia, Pa, USA, German edition, 1991. · Zbl 0746.16001 |
| [7] | F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, vol. 13 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1974. · Zbl 0301.16001 |
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.
