## The polarity along an element in rings.(English)Zbl 07731156

The authors study various notions of generalized inverses in rings. An element $$x$$ is Drazin inverse to $$a$$ if $$x$$ commutes with $$a$$, $$xax=x$$ and there is an integer $$n$$ such that $$xa^{n+1}=a^n$$. An element $$a$$ is regular if there is $$x$$ such that $$axa=a$$. Such an $$x$$ is called an inner inverse to $$a$$. Another generalisation is the inverse along an element $$d$$, and finally the Moore Penrose inverse of an element. Here, $$a$$ is invertible along $$d$$ if there is $$y$$ in $$dR\cap Rd$$ and $$yad=d=day$$.
Parallel to this there is the concept of a polar element and various generalisations thereof. An element $$a$$ is polar if there is an idempotent $$p$$ commuting with $$a$$ such that $$a(1-p)$$ is nilpotent, and $$a+(1-p)$$ is truly invertible. The authors introduces the concept of being polar along an element $$d$$ by the following condition. $$a$$ is polar along $$d$$ if there is an idempotent $$q$$ commuting with $$ad$$ such that $$dq=d$$ and $$ad+(1-q)$$ is truly invertible.
The first link comes from the statement that $$a$$ is Drazin invertible iff it is polar. They then show that $$a$$ is invertible along $$d$$ iff $$a$$ is polar along $$d$$. Further more technical conditions of this kind are shown.

### MSC:

 16U60 Units, groups of units (associative rings and algebras) 16W10 Rings with involution; Lie, Jordan and other nonassociative structures

### Keywords:

generalised inverses; polar elements
Full Text:

### References:

 [1] O. M. Baksalary and G. Trenkler, “Core inverse of matrices”, Linear Multilinear Algebra 58:5-6 (2010), 681-697. · Zbl 1202.15009 · doi:10.1080/03081080902778222 [2] J. Benítez and E. Boasso, “The inverse along an element in rings”, Electron. J. Linear Algebra 31 (2016), 572-592. · Zbl 1351.15004 · doi:10.13001/1081-3810.3113 [3] J. Benítez and E. Boasso, “The inverse along an element in rings with an involution, Banach algebras and $C^\ast$-algebras”, Linear Multilinear Algebra 65:2 (2017), 284-299. · Zbl 1361.15004 · doi:10.1080/03081087.2016.1183559 [4] K. P. S. Bhaskara Rao, The theory of generalized inverses over commutative rings, Algebra, Logic and Applications 17, Taylor and Francis, London, 2002. [5] D. S. Djordjević and V. Rakočević, Lectures on generalized inverses, University of Niš, 2008. · Zbl 1419.47001 [6] M. P. Drazin, “Pseudo-inverses in associative rings and semigroups”, Amer. Math. Monthly 65 (1958), 506-514. · Zbl 0083.02901 · doi:10.2307/2308576 [7] R. E. Hartwig and J. Shoaf, “Group inverses and Drazin inverses of bidiagonal and triangular Toeplitz matrices”, J. Austral. Math. Soc. Ser. A 24:1 (1977), 10-34. · Zbl 0372.15003 · doi:10.1017/s1446788700020036 [8] J. J. Koliha, “A generalized Drazin inverse”, Glasgow Math. J. 38:3 (1996), 367-381. · Zbl 0897.47002 · doi:10.1017/S0017089500031803 [9] J. J. Koliha, D. Djordjević, and D. Cvetković, “Moore-Penrose inverse in rings with involution”, Linear Algebra Appl. 426:2-3 (2007), 371-381. · Zbl 1130.46032 · doi:10.1016/j.laa.2007.05.012 [10] J. J. Koliha and P. Patricio, “Elements of rings with equal spectral idempotents”, J. Aust. Math. Soc. 72:1 (2002), 137-152. · Zbl 0999.16025 · doi:10.1017/S1446788700003657 [11] X. Mary, “On generalized inverses and Green’s relations”, Linear Algebra Appl. 434:8 (2011), 1836-1844. · Zbl 1219.15007 · doi:10.1016/j.laa.2010.11.045 [12] X. Mary and P. Patrício, “Generalized inverses modulo $\mathcal{H}$ in semigroups and rings”, Linear Multilinear Algebra 61:8 (2013), 1130-1135. · Zbl 1383.15005 · doi:10.1080/03081087.2012.731054 [13] R. Penrose, “A generalized inverse for matrices”, Proc. Cambridge Philos. Soc. 51 (1955), 406-413. · Zbl 0065.24603 · doi:10.1017/S0305004100030401 [14] D. S. Rakić, N. Dinčić, and D. S. Djordjević, “Group, Moore-Penrose, core and dual core inverse in rings with involution”, Linear Algebra Appl. 463 (2014), 115-133. · Zbl 1297.15006 · doi:10.1016/j.laa.2014.09.003 [15] S. Roch and B. Silbermann, “Continuity of generalized inverses in Banach algebras”, Studia Math. 136:3 (1999), 197-227. · Zbl 0962.47002 [16] S. Xu and J. Chen, “The Moore-Penrose inverse in rings with involution”, Filomat 33:18 (2019), 5791-5802. · Zbl 1513.16052 · doi:10.2298/fil1918791x [17] H. Zhu, “Further results on several types of generalized inverses”, Comm. Algebra 46:8 (2018), 3388-3396. · Zbl 1390.16035 · doi:10.1080/00927872.2017.1412450 [18] H. Zhu and J. Chen, “Additive and product properties of Drazin inverses of elements in a ring”, Bull. Malays. Math. Sci. Soc. 40:1 (2017), 259-278. · Zbl 1361.15008 · doi:10.1007/s40840-016-0318-2 [19] H. Zhu, J. Chen, and P. Patrício, “Further results on the inverse along an element in semigroups and rings”, Linear Multilinear Algebra 64:3 (2016), 393-403. · Zbl 1338.15014 · doi:10.1080/03081087.2015.1043716 [20] H. Zhu, J. Chen, P. Patrício, and X. Mary, “Centralizer’s applications to the inverse along an element”, Appl. Math. Comput. 315 (2017), 27-33. · Zbl 1426.15005 · doi:10.1016/j.amc.2017.07.046 [21] H. Zhu, L. Wu, and D. Mosić, “One-sided $w$-core inverses in rings with an involution”, Linear Multilinear Algebra 71:4 (2023), 528-544. · Zbl 1519.16034 · doi:10.1080/03081087.2022.2035308 [22] H. Zhu, X. Zhang, and J. Chen, “Centralizers and their applications to generalized inverses”, Linear Algebra Appl. 458 (2014), 291-300 · Zbl 1294.15006 · doi:10.1016/j.laa.2014.06.015
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