Reflexivity with maximal ideal axes

Reflexivity of rings via nilpotent elements

“…Let R = A/I and let x = X + I, y = Y + I ∈ R. Then in R, x 3 = 0, y 3 = 0, xy = 0, yx 2 = 0, y 2 x = 0. In [1,Example 2.3], xRy = 0, yRx = 0, and idempotents in R are 0 and 1. Hence for any r ∈ nil(R) and e 2 = e ∈ R, rRe = 0 implies eRr = 0.…”

“…In [17], a right ideal I of R is said to be reflexive if aRb ⊆ I implies bRa ⊆ I for any a, b ∈ R. A ring R is called reflexive if 0 is a reflexive ideal of R. Reversible rings are reflexive by [14,Proposition 2.2]. In [19], R is said to be a weakly reflexive ring if aRb = 0 implies bRa ⊆ nil(R) for any a, b ∈ R. In [13], a ring R is said to be nil-reflexive if aRb ⊆ nil(R) implies that bRa ⊆ nil(R) for any a, b ∈ R. In [1], R is called a reflexivity with maximal ideal axis ring (an RM ring, for short) if for a maximal ideal M and for any a, b ∈ R, aM b = 0 implies bM a = 0; similarly, R has reflexivity with maximal ideal axis on idempotents (simply, RMI ) if eM f = 0 for any idempotents e, f and a maximal ideal of M yields f M e = 0. In [15], R has reflexive-idempotents-property (simply, RIP) if eRf = 0 for any idempotents e, f yields f Re = 0.…”

Reflexivity of Rings via Nilpotent Elements

“…And in [14], R is said to be a weakly reflexive ring if aRb = 0 implies bRa ⊆ nil(R) for any a, b ∈ R. In [8], a ring R is said to be nil-reflexive if aRb ⊆ nil(R) implies that bRa ⊆ nil(R) for any a, b ∈ R. Let R be a ring. In [1], R is called a reflexivity with maximal ideal axis ring, in short, an RM ring if aM b = 0 for a maximal ideal M and for any a, b ∈ R, then bM a = 0, similarly, R has reflexivity with maximal ideal axis on idempotents, simply, RMI, if eM f = 0 for any idempotents e, f and a maximal ideal of M , then f M e = 0. In [10], R has reflexive-idempotents-property, simply, RIP, if eRf = 0 for any idempotents e, f , then f Re = 0, A left ideal I is called idempotent reflexive [6] if aRe ⊆ I implies eRa ⊆ I for a, e 2 = e ∈ R.…”

“…(1) Let F be a field and consider the ring R = is quasi-Armendariz by[5, Corollary 3.15]. However, R is not left N-reflexive.…”