In this subsection, we will assume that R is a prime ring that H is a generalization of R, that L is a noncentral Lie idyllic of R, and that 0???R. If s, t ? 0 and n > 0, then for all u in L, aus(H(u)n)u(t) must equal 0. All of them fit neatly inside the specified ranges of integers. H(x) = bx for every x ? R if and only if s = 0 and b = 0; this is not the case if R satisfy s4, the normal uniqueness in four variables. In this context, b stands for the Utumi quotient ring of the right side of R. The set R is said to meet the S4 condition if and only if the function H(x) is equal to bx. Once s is equal to zero, H(x) equals bx for every x in R. It follows that H = 0 if and only if s > 0 and R does not meet S4 the requirement.
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