In this paper, we first give some characterizations of e-symmetric rings. We prove that R is an e-symmetric ring if and only if a1a2a3 = 0 implies that aσ(1)aσ(2)aσ(3)e = 0, where σ is any transformation of {1, 2, 3}. With the help of the Bott–Duffin inverse. | Turk J Math (2018) 42: 2389 – 2399 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Some properties of e-symmetric rings Fanyun MENG∗, Junchao WEI, School of Mathematics, Yangzhou University, Yangzhou, . China Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we first give some characterizations of e -symmetric rings. We prove that R is an e -symmetric ring if and only if a1 a2 a3 = 0 implies that aσ(1) aσ(2) aσ(3) e = 0 , where σ is any transformation of {1, 2, 3} . With the help of the Bott–Duffin inverse, we show that for e ∈ M El (R) , R is an e -symmetric ring if and only if for any a ∈ R and g ∈ E(R) , if a has a Bott–Duffin (e, g) -inverse, then g = eg . Using the solution of the equation axe = c , we show that for e ∈ M El (R) , R is an e -symmetric ring if and only if for any a, c ∈ R , if the equation axe = c has a solution, then c = ec . Next, we study the properties of e -symmetric ∗ -rings. Finally we discuss when the upper triangular matrix ring T2 (R) (resp. T3 (R, I) ) becomes an e -symmetric ring, where e ∈ E(T2 (R)) (resp. e ∈ E(T3 (R, I)) ). Key words: e -Symmetric ring, ∗ -ring, left semicentral, left min-abel ring, Bott–Duffin inverse, upper triangular matrix ring 1. Introduction Throughout this paper, all rings are associative with unity. For a ring R , T2 (R) denotes the 2 × 2 upper triangular matrix ring over R , and E(R) , U (R) , Z(R) , and N (R) denote the set of all idempotents, the set of all invertible elements, the center of R , and the set of all nilpotent elements of R , respectively. An element e ∈ E(R) is called left minimal idempotent of R if Re is a minimal left ideal of R . Write M El (R) to denote the set of all left minimal idempotents of R . An idempotent e of a ring R is called left (right) semicentral ae = eae (ea = eae ) for each a ∈ R . A ring R is called (strongly) left .
