Let \(D\) be a division ring and \(K\) a subfield of \(D\) which is not necessarily contained in the center \(F\) of \(D\). We study the structure of \(D\) under the condition of left algebraicity of certain subsets of \(D\) over \(K\). Among others, it is proved that if \(D^*\) contains a noncentral normal subgroup which is left algebraic over \(K\) of bounded degree \(d\), then \([D:F]\le d^2\). In case \(K=F\), the obtained results show that if either all additive commutators or all multiplicative commutators with respect to a noncentral subnormal subgroup of \(D^*\) are algebraic of bounded degree \(d\) over \(f\), then \([D:F]\le d^2\).

让\（d \）是除环和\（K \）的子场\（d \）包括不一定包含在中心\（F \）的\（d \） 。我们研究在\（K \）上\（D \）的某些子集的左代数性的情况下\（D \）的结构。其中，证明了如果\（D ^ * \）包含非中心法子子集，且该子集在有界度\（d \）的\（K \）上代数，则\（[D：F] \ le d ^ 2 \）。如果\（K = F \），所获得的结果表明，如果只将所有添加剂换向器或所有乘法换向器相对于一个非中心低于正常的亚组\（d ^ * \）是代数界度\（d \）超过\（F \） ，然后\（ [D：F] \ le d ^ 2 \）。