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Unit-regularity and representability for semiartinian \(*\)-regular rings. (English) Zbl 07177879

A ring \(R\) is called a \(*\)-ring if \(R\) is endowed with an involution \(r\mapsto r^*\). A \(*\)-ring is called \(*\)-regular if it is (von Neumann) regular and \(rr^* =0\) only for \(r=0\). The main results of the paper show that: (1) Any semiartinian \(*\)-regular ring is unit-regular; (2) If \(R\) is a subdirectly irreducible \(*\)-regular ring with non-zero socle, then it admits a representation within some inner product space.

MSC:

16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
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[1] Baccella, G.; Spinosa, L., \(K_0\) of semiartinian von Neumann regular rings. Direct finiteness versusunit-regularity, Algebr. Represent. Theory 20 (2017), 1189-1213 · Zbl 1376.16003 · doi:10.1007/s10468-017-9682-3
[2] Berberian, S. K., Baer *-rings, Springer, Grundlehren 195, Berlin, 1972 · Zbl 0242.16008
[3] Goodearl, K. R., Von Neumann Regular Rings, 2nd ed., Krieger, Malabar, 1991 · Zbl 0749.16001
[4] Gross, H., Quadratic Forms in Infinite Dimensional Vector spaces, Birkhäuser, Basel, 1979 · Zbl 0413.10013
[5] Handelman, D., Finite Rickart \(C^*\)-algebras and their properties, Adv. in Math. Suppl. Stud., vol. 4, Academic Press, New York-London, Studies in analysis ed., 1979, pp. 171-196 · Zbl 0511.46054
[6] Herrmann, C., Varieties of \(*\)-regular rings, http://arxiv.org/abs/1904.04505 · Zbl 1476.16008
[7] Herrmann, C., On the equational theory of projection lattices of finitevon Neumann factors, J. Symbolic Logic 75 (3) (2010), 1102-1110 · Zbl 1205.06005 · doi:10.2178/jsl/1278682219
[8] Herrmann, C., Direct finiteness of representable regular \(*\)-rings, Algebra Universalis 80 (1) (2019), 5 pp., http://arxiv.org/abs/1904.04505 · Zbl 1448.16012
[9] Herrmann, C.; Semenova, M. V., Rings of quotients of finite \(AW^*\)-algebras. Representation and algebraic approximation, Algebra Logika 53 (4) (2014), 466-504, 550-551, (Russian), translation inAlgebra Logic 53 (2014), no. 4, 298-322 · Zbl 1323.46032 · doi:10.1007/s10469-014-9292-7
[10] Herrmann, C.; Semenova, M. V., Linear representations of regular rings and complemented modular lattices with involution, Acta Sci. Math. (Szeged) 82 (3-4) (2016), 395-442 · Zbl 1399.06023 · doi:10.14232/actasm-015-283-5
[11] Jacobson, N., Structure of Rings, AMS Col. Publ. XXXVII, Amer. Math. Soc., Providence, RI, 1956 · Zbl 0073.02002
[12] Micol, F., On representability of \(\ast \)-regular rings and modular ortholattices, Ph.D. thesis, TU Darmstadt, January 2003, http://elib.tu-darmstadt.de/diss/000303/diss.pdf · Zbl 1085.16505
[13] Wehrung, F., A uniform refinement property for congruence lattices, Proc. Amer. Math. Soc. 127 (1999), 363-370 · Zbl 0902.06006 · doi:10.1090/S0002-9939-99-04558-X
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