tailieunhanh - Quasinilpotents in rings and their applications
An element a of an associative ring R is said to be quasinilpotent if 1 − ax is invertible for every x ∈ R with xa = ax. Nilpotents and elements in the Jacobson radical of a ring are well-known examples of quasinilpotents. In this paper, properties and examples of quasinilpotents in a ring are provided, and the set of quasinilpotents is applied to characterize rings with some certain properties. | Turk J Math (2018) 42: 2854 – 2862 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Quasinilpotents in rings and their applications Jian CUI∗ Department of Mathematics, Anhui Normal University, Wuhu, . China Received: • Accepted/Published Online: • Final Version: Abstract: An element a of an associative ring R is said to be quasinilpotent if 1 − ax is invertible for every x ∈ R with xa = ax . Nilpotents and elements in the Jacobson radical of a ring are well-known examples of quasinilpotents. In this paper, properties and examples of quasinilpotents in a ring are provided, and the set of quasinilpotents is applied to characterize rings with some certain properties. Key words: Quasinilpotent, nilpotent, idempotent, local ring, Boolean ring 1. Introduction Rings are associative with identity. Let R be a ring. The symbols U (R) , Id(R), and Rnil stand for the sets of all units, all idempotents, and all nilpotents of R , respectively. The commutant of a ∈ R is defined by comm R (a) = {x ∈ R | ax = xa} (if there is no ambiguity, we simply use comm (a) for short). For an integer n ≥ 1, we write Mn (R) for the n × n matrix ring over R whose identity element we write as In or I . The intersection of all maximal left (right) ideals of R is said to be the Jacobson radical of R , which is denoted by J(R). As is well known, J(R) = {a ∈ R | 1 − ax ∈ U (R) for all x ∈ R} . Due to Harte [10], an element a ∈ R is called quasinilpotent if 1 − ax ∈ U (R) for every x ∈ comm(a); the set of all quasinilpotents of R is denoted by Rqnil . It is clear that both Rnil and J(R) are contained in Rqnil . It is worth noting that quasinilpotents play an important role in a Banach algebra A. According to [9], Aqnil = {a ∈ A | lim ∥an ∥1/n = 0} = {a ∈ A | x − a ∈ U (A) for all nonzero complex x} . By means of n→∞ quasinilpotents, some interesting concepts are introduced, such as .
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