tailieunhanh - Certain strongly clean matrices over local rings

We are concerned about various strongly clean properties of a kind of matrix subrings L(s)(R) over a local ring R. Let R be a local ring, and let s ∈ C(R). We prove that A ∈ L(s)(R) is strongly clean if and only if A or I2−A is invertible, or A is similar to a diagonal matrix in L(s)(R). | Turk J Math (2018) 42: 2296 – 2303 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Certain strongly clean matrices over local rings 1 Tuğçe Pekacar ÇALCI1,∗, Huanyin CHEN,2 Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey 2 Department of Mathematics, Hangzhou Normal University, Hangzhou, . China Received: • Accepted/Published Online: • Final Version: Abstract: We are concerned about various strongly clean properties of a kind of matrix subrings L(s) (R) over a local ring R . Let R be a local ring, and let s ∈ C(R) . We prove that A ∈ L(s) (R) is strongly clean if and only if A or I2 − A is invertible, or A is similar to a diagonal matrix in L(s) (R) . Furthermore, we prove that A ∈ L(s) (R) is quasipolar ( ) λ 0 if and only if A ∈ GL2 (R) or A ∈ L(s) (R)qnil , or A is similar to a diagonal matrix in L(s) (R) , where 0 µ λ ∈ J(R) , µ ∈ U (R) or λ ∈ U (R) , µ ∈ J(R) , and lµ − rλ , lλ − rµ are injective. Pseudopolarity of such matrix subrings is also obtained. Key words: Matrix ring, strongly clean matrix, quasipolar matrix 1. Introduction Throughout, all rings are associative with an identity. An element a in a ring R is strongly clean provided that it is the sum of an idempotent and a unit that commute. The commutant of a ∈ R is defined by comm(a) = {x ∈ R | xa = ax}. Set Rqnil = {a ∈ R | 1 + ax ∈ U (R) for every x ∈ comm(a)} . We say a ∈ R is quasinilpotent if a ∈ Rqnil . The double commutant of a ∈ R is defined by comm2 (a) = {x ∈ R | xy = yx for all y ∈ comm(a)} . In [6], an element a in a ring R is called quasipolar if for any a ∈ R there exists e2 = e ∈ comm2 (a) such that a + e ∈ U (R) and ae ∈ Rqnil . As is well known, an element a ∈ R is quasipolar if and only if it has generalized Drazin inverse, . there exists b ∈ comm2 (a) such that b = b2 a, a − a2 b ∈ Rqnil (see [6]). Following [8], an element a in a