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**ADS modules.**
*(English)*
Zbl 1256.16005

The notion of modules with the ‘absolute direct summand property’ (ADS, for short) was introduced by L. Fuchs [in Infinite Abelian groups. Vol. I. Pure and Applied Mathematics 36. New York-London: Academic Press (1970; Zbl 0209.05503)]. Fuchs called a right module \(M\) right ADS if for every decomposition \(M=S\oplus T\) of \(M\) and every complement \(C\) of \(S\) we have \(M=S\oplus C\). For example, if \(R\) is an Abelian ring then \(R\) as a right \(R\)-module is ADS. Every right quasi-continuous module is ADS, but the converse need not be true. If a right ADS module is also right CS, then it is right quasi-continuous.

The paper under review shows that if \(R\) is a simple ring such that \(R_R\) is ADS, then either \(R\) is right self-injective or indecomposable as a right \(R\)-module. W. D. Burgess and R. Raphael [in Ring theory. Proceedings of the biennial Ohio State-Denison mathematics conference, 1992. Singapore: World Scientific. 137-148 (1993; Zbl 0853.16008)] claimed that an example can be constructed of a finite dimensional module over a finite dimensional algebra which has no ADS hull. This paper shows that under certain conditions such an ADS hull does exist. A right module \(M\) is called completely ADS if each of its subfactors is ADS. Among other results it is shown that if \(M\) is a semiperfect module with a completely ADS projective cover, then \(M=S\oplus T\) where \(S\) is semisimple and \(T\) is a sum of local modules.

The paper under review shows that if \(R\) is a simple ring such that \(R_R\) is ADS, then either \(R\) is right self-injective or indecomposable as a right \(R\)-module. W. D. Burgess and R. Raphael [in Ring theory. Proceedings of the biennial Ohio State-Denison mathematics conference, 1992. Singapore: World Scientific. 137-148 (1993; Zbl 0853.16008)] claimed that an example can be constructed of a finite dimensional module over a finite dimensional algebra which has no ADS hull. This paper shows that under certain conditions such an ADS hull does exist. A right module \(M\) is called completely ADS if each of its subfactors is ADS. Among other results it is shown that if \(M\) is a semiperfect module with a completely ADS projective cover, then \(M=S\oplus T\) where \(S\) is semisimple and \(T\) is a sum of local modules.

Reviewer: Ashish K. Srivastava (Saint Louis)

### MSC:

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

16D50 | Injective modules, self-injective associative rings |

16D80 | Other classes of modules and ideals in associative algebras |

### Keywords:

ADS modules; injective modules; quasi-continuous modules; direct sums; absolute direct summand property; ADS hulls### References:

[1] | Burgess, W. D.; Raphael, R., On modules with the absolute direct summand property, (Ring Theory. Ring Theory, Granville, OH, 1992 (1993), World Sci. Publ.: World Sci. Publ. River Edge, NJ), 137-148 · Zbl 0853.16008 |

[2] | Fuchs, L., Infinite Abelian Groups, vol. I, Pure Appl. Math., Ser. Monogr. Textb., vol. 36 (1970), Academic Press: Academic Press New York, San Francisco, London · Zbl 0209.05503 |

[3] | Goodearl, K. R., Von Neumann Regular Rings (1991), Krieger Publishing Company: Krieger Publishing Company Malabar, FL · Zbl 0749.16001 |

[4] | Goel, V. K.; Jain, S. K., \(π\)-injective modules and rings whose cyclics are \(π\)-injective, Comm. Algebra, 6, 59-73 (1978) · Zbl 0368.16010 |

[5] | Kasch, F., Modules and Rings (1982), Academic Press · Zbl 0523.16001 |

[6] | Mohammed, S. H.; Müller, B. J., Continuous and Discrete Modules, London Math. Soc. Lecture Note Ser., vol. 147 (1990), Cambridge University Press: Cambridge University Press New York · Zbl 0701.16001 |

[7] | Wisbauer, R., Foundations of Module and Ring Theory (1991), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers Reading · Zbl 0746.16001 |

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