Is a full ritual bath (ghusl) required, or only washing the private parts, when sexual desire occurs?

If lust is aroused without the emission of semen, then ghusl (ritual bath) is not obligatory. Rather, only wudu' (ablution) is required, along with washing the male organ and whatever was affected by madhy (pre-seminal fluid) if madhy was discharged. However, if the arousal of lust led to the emission of semen, then a full ghusl is obligatory.