Let R be a unitary prime ring with the maximal left ring of quotients $Q_{ml}(R)$. We completely characterize the forms of additive maps $F_1,F_2,G_1,G_2:R\to Q_{ml}(R)$ such that $F_1(x)y+F_2(y)x+x{G}_1(y)+y{G}_2(x)=0$whenever $x,y\in R$ satisfy xy = 0 = yx under the assumptions that R contains a nontrivial idempotent and characteristic of R is not 2. This partially generalizes a result of T.K. Lee in [Bi-additive maps of ζ-Lie product type vanishing on zero products of XY and Y X. Comm Algebra. 2017;45(8):3449–3467].

中文翻译：

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XY 和 YX 的零积上消失的 2 次函数恒等式

令R为具有最大左环商的酉素环${问}_{米升}(R)$. 我们完全描述了加法映射的形式$F_{\mathrm{1个}},F_{\mathrm{2个}},G_{\mathrm{1个}},G_{\mathrm{2个}}:R\to {问}_{米升}(R)$这样$F_{\mathrm{1个}}(X)是+F_{\mathrm{2个}}(是)X+X{G}_{\mathrm{1个}}(是)+是G_{\mathrm{2个}}(X)=0$每当$X,是\in R$满足xy  = 0 =  yx假设R包含非平凡的幂等性且R的特征不为 2。这部分概括了 TK Lee 在 [Bi-additive maps of ζ -Lie product type vanishing on zero products of XY and YX . 通信代数。2017;45(8):3449–3467]。