Geodesic
Conrad’s Partial Order on P.Q.-Baer *-Rings
Discussiones Mathematicae. General Algebra and Applications, Tome 38 (2018) no. 2, pp. 207-219.

Voir la notice de l'article provenant de la source Library of Science

We prove that a p.q.-Baer *-ring forms a pseudo lattice with Conrad’s partial order and also characterize p.q.-Baer *-rings which are lattices. The initial segments of a p.q.-Baer *-ring with the Conrad’s partial order are shown to be an orthomodular posets.
Keywords: Conrad’s partial order, p.q.-Baer *-ring, central cover, orthomodular set 
@article{DMGAA_2018_38_2_a3,
     author = {Khairnar, Anil and Waphare, B.N.},
     title = {Conrad{\textquoteright}s {Partial} {Order} on {P.Q.-Baer} {*-Rings}},
     journal = {Discussiones Mathematicae. General Algebra and Applications},
     pages = {207--219},
     publisher = {mathdoc},
     volume = {38},
     number = {2},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGAA_2018_38_2_a3/}
}
TY  - JOUR
AU  - Khairnar, Anil
AU  - Waphare, B.N.
TI  - Conrad’s Partial Order on P.Q.-Baer *-Rings
JO  - Discussiones Mathematicae. General Algebra and Applications
PY  - 2018
SP  - 207
EP  - 219
VL  - 38
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGAA_2018_38_2_a3/
LA  - en
ID  - DMGAA_2018_38_2_a3
ER  - 
%0 Journal Article
%A Khairnar, Anil
%A Waphare, B.N.
%T Conrad’s Partial Order on P.Q.-Baer *-Rings
%J Discussiones Mathematicae. General Algebra and Applications
%D 2018
%P 207-219
%V 38
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGAA_2018_38_2_a3/
%G en
%F DMGAA_2018_38_2_a3
Khairnar, Anil; Waphare, B.N. Conrad’s Partial Order on P.Q.-Baer *-Rings. Discussiones Mathematicae. General Algebra and Applications, Tome 38 (2018) no. 2, pp. 207-219. http://geodesic.mathdoc.fr/item/DMGAA_2018_38_2_a3/

[1] A. Abian, Direct product decomposition of commutative semisimple rings, Proc. Amer. Math. Soc. 24 (1970) 502–507. doi:10.2307/2037396

[2] J.K. Baksalary and S.K. Mitra, Left-star and right-star partial ordering, Linear Algebra Appl. 149 (1991) 73–89. doi:10.1016/0024-3795(91)90326-R

[3] S.K. Berberian, Baer * Rings, Grundlehren Math.Wiss. Band 195. Vol. 296 (Berlin, Springer, 1972). doi:10.1007/978-3-642-15071-5

[4] G.F. Birkenmeier, J.K. Park and S.T. Rizvi, Principally quasi-Baer rings hulls, Advances in Ring Theory Trends in Mathematics, 47–61 (Birkhäuser Basel, 2010). doi:10.1007/978-3-0346-0286-0\_4

[5] G.F. Birkenmeier, J.K. Park and S.R. Tariq, Extensions of Rings and Modules (New York, Birkhäuser, 2013). doi:10.1007/978-0-387-92716-9

[6] W.D. Burgess and R. Raphael, On Conrad’s partial order relation on semiprime rings and on semigroups, Semigroup Forum 16 (1978) 133–140. http://eudml.org/doc/134282

[7] J. Cīrulis, Quasi-orthomodular posets and weak BCK-algebras, Order 31 (2014) 403–419. doi:10.1007/s11083-013-9309-1

[8] P.F. Conrad, The hulls of semiprime rings, Austral. Math. Soc. 12 (1975) 311–314. doi:10.1017/S0004972700023911

[9] G. Dolinar and J. Marovt, Star partial order on B (H), Linear Algebra Appl. 434 (2011) 319–326. doi:10.1016/j.laa.2010.08.023

[10] G. Dolinar, B. Kuzma and J. Marovt, A note on partial orders of Hartwig, Mitsch, and Šemrl, Appl. Math. and Comp. 270 (2015) 711–713. doi:10.1016/j.amc.2015.08.066

[11] M.P. Drazin, Natural structure on semigroup with involution, Bull. Amer. Math. Soc. 84 (1978) 139–141. https://projecteuclid.org/euclid.bams/1183540393

[12] R.E. Hartwig, How to partially order regular elements, Math. Japon. 25 (1980) 1–13.

[13] R.E. Hartwig, Pseudo lattice properties of the star-orthogonal partial ordering for star-regular rings., Proc. Amer. Math. Soc. 77 (1979) 299–303. doi:10.2307/2042174

[14] C.Y. Hong, N.K. Kim, T.K. Kwak, Ore extension of Baer and PP rings, J. Pure Appl. Algebra 151 (2000) 215–226. doi:10.1016/S0022-4049(99)00020-1

[15] M.F. Janowitz, On the * -order for Rickart * -rings, Algebra Universalis 16 (1983) 360–369. doi:10.1007/BF01191791

[16] I. Kaplansky, Rings of Operators (W.A. Benjamin, Inc., New York-Amsterdam, 1968).

[17] A. Khairnar and B.N. Waphare, Unitification of weakly p.q.-Baer * -rings, Southeast Asian Bull. Math. (to appear). arXiv:1612.01681

[18] A. Khairnar and B.N. Waphare, Order properties of generalized projections, Linear Multilinear Algebra 65 (2017) 1446–1461. doi:10.1080/03081087.2016.1242554

[19] A. Khairnar and B.N. Waphare, Generalized Projections in Z n, AKCE Int. J. Graphs Comb. doi:10.1016/j.akcej.2018.01.010

[20] A. Khairnar and B.N. Waphare, A Sheaf Representation of Principally Quasi-Baer * -Rings, Algebr. Represent. Theory. https://doi.org/10.1007/s10468-017-9758-0

[21] J.Y. Kim, On reflexive principally quasi-Baer rings, Korean J. Math. 17 (2009) 233–236.

[22] I. Krēmere, Left-star order structure of Rickart * -ring, Linear Multilinear Algebra 64 (2016) 341–352. doi:10.1080/03081087.2015.1040369

[23] F.W. Levi, Ordered groups, Proc. Indian Acad. Sci. A 16 (1942) 256–263.

[24] S. Maeda, On the lattice of projections of a Baer * -ring, J. Sci. Hiroshima Univ. Ser. A 22 (1958) 75–88.

[25] P. Šemrl, Automorphisms of B (H) with respect to minus partial order, J. Math. Anal. Appl. 369 (2010) 205–213. doi:10.1016/j.jmaa.2010.02.059

[26] B.N. Waphare and Anil Khairnar, Semi-Baer modules, J. Algebra Appl. 14 (2015) 1550145 (12 pages). doi:10.1142/S0219498815501455