Baer and quasi-Baer modules. (English) Zbl 1072.16007

Rizvi, S. Tariq; Roman, Cosmin S.

Commun. Algebra 32, No. 1, 103-123 (2004).

The notions of Baer and quasi-Baer properties in a general module theoretic setting are introduced. A module \(M_R\) is called (quasi-)Baer if the right annihilator of a left (two-sided) ideal of \(\text{End}(M_R)\) is a direct summand. It is shown that a direct summand of a (quasi-)Baer module is (quasi-)Baer and every finitely generated Abelian group is Baer exactly if it is semisimple or torsion-free.
A \(\mathcal K\)-nonsingular module is defined to study (quasi-)Baer modules. A module \(M\) is called \(\mathcal K\)-nonsingular if, for all \(\varphi\in\text{End}(M_R)\), \(\text{Ker}(\varphi)\) is essential in \(M_R\), then \(\varphi=0\). Close connections to the (FI-)extending property are investigated and it is shown that a module \(M\) is (quasi-)Baer and (FI-)\(\mathcal K\)-cononsingular if and only if it is (FI-)extending and (FI-)extending \(\mathcal K\)-nonsingular. Also it is proved that an arbitrary direct sum of mutually subisomorphic quasi-Baer modules is quasi-Baer and every free (projective) module over a quasi-Baer ring is a quasi-Baer module. Furthermore, among other results, it is shown that the endomorphism ring of a (quasi-)Baer module is a (quasi-)Baer ring, while the converse is not true in general. Interesting applications of the results are provided.

Reviewer: J. K. Park (Pusan)

MSC:

16D80	Other classes of 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)
16S50	Endomorphism rings; matrix rings

Keywords:

Baer rings; Baer modules; idempotents; right annihilators; direct summands; quasi-Baer modules; FI-extending property; direct sums; quasi-Baer rings; endomorphism rings

References:

[1]	DOI: 10.1007/978-3-642-15071-5 · doi:10.1007/978-3-642-15071-5
[2]	DOI: 10.1006/jabr.2000.8328 · Zbl 0964.16031 · doi:10.1006/jabr.2000.8328
[3]	DOI: 10.1016/S0022-4049(99)00164-4 · Zbl 0947.16018 · doi:10.1016/S0022-4049(99)00164-4
[4]	DOI: 10.1017/S0004972700022000 · Zbl 0952.16009 · doi:10.1017/S0004972700022000
[5]	DOI: 10.1090/conm/259/04088 · doi:10.1090/conm/259/04088
[6]	DOI: 10.1080/00927870008826857 · Zbl 0954.16002 · doi:10.1080/00927870008826857
[7]	Birkenmeier, G. F., Kim, J. Y. and Park, J. K. Semicentral reduced algebras. International Symposium on Ring Theory, Edited by: Birkenmeier, G. F., Park, J. K. and Park, Y. S. pp.67–84. Boston: Birkhäuser. Trends in Math · Zbl 0987.16010
[8]	DOI: 10.1081/AGB-100001530 · Zbl 0991.16005 · doi:10.1081/AGB-100001530
[9]	DOI: 10.1080/00927870209342387 · doi:10.1080/00927870209342387
[10]	DOI: 10.1081/AGB-120013220 · Zbl 1005.16005 · doi:10.1081/AGB-120013220
[11]	DOI: 10.1216/rmjm/1181070024 · Zbl 1035.16024 · doi:10.1216/rmjm/1181070024
[12]	DOI: 10.1112/jlms/s2-21.3.434 · Zbl 0432.16017 · doi:10.1112/jlms/s2-21.3.434
[13]	DOI: 10.1215/S0012-7094-67-03446-1 · Zbl 0204.04502 · doi:10.1215/S0012-7094-67-03446-1
[14]	Dung Nguyen Viet, Pitman Research Notes in Mathematics Series, 314, in: Extending Modules (1994)
[15]	Fuchs L., Pure and Applied Mathematics Series 36, in: Infinite Abelian Groups (1970)
[16]	Kaplansky I., Mathematics Lecture Note Series, in: Rings of Operators (1968)
[17]	DOI: 10.1017/S000497270002877X · Zbl 0719.16004 · doi:10.1017/S000497270002877X
[18]	Kim J. Y., Math. Japonica 45 (2) pp 311– (1997)
[19]	DOI: 10.1007/978-1-4612-0525-8 · doi:10.1007/978-1-4612-0525-8
[20]	DOI: 10.1090/S0002-9939-1974-0357466-X · doi:10.1090/S0002-9939-1974-0357466-X
[21]	DOI: 10.1007/BF01111594 · Zbl 0215.38102 · doi:10.1007/BF01111594
[22]	DOI: 10.1215/S0012-7094-70-03718-X · Zbl 0219.16010 · doi:10.1215/S0012-7094-70-03718-X
[23]	DOI: 10.1007/978-3-642-66066-5 · doi:10.1007/978-3-642-66066-5
[24]	DOI: 10.1080/00927878608823297 · Zbl 0592.13008 · doi:10.1080/00927878608823297
[25]	Wisbauer R., Pitman Monographs and Surveys in Pure and Applied Mathematics, 81. Harlow: Longman, in: Modules and Algebras: Bimodule Structure and Group Actions on Algebras (1996)
[26]	DOI: 10.1090/S0002-9947-1972-0286843-3 · doi:10.1090/S0002-9947-1972-0286843-3