Some properties of e -symmetric rings

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, we show that for e ∈ MEl(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 ∈ MEl(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))).