NOVELTY - A transition-metal compound containing nickel and metal (M) other than lithium and nickel are mixed in a molar ratio of a:(1-a) with lithium carbonate in a preset ratio. The temperature of the mixture is made to reach a preset temperature range while repeatedly raising and lowering the temperature. The transition-metal compound is then reacted with lithium carbonate at the preset temperature range, to obtain a lithium-containing transition-metal oxide.    USE - Production of lithium-containing transition-metal oxide useful as positive-electrode active material for non-aqueous electrolyte batteries e.g. lithium secondary batteries. Uses include but are not limited to notebook personal computers, cellular phones, digital still cameras, power storage, and electric vehicles.    ADVANTAGE - The method reduces the problems arising from the use of a nickel-containing transition-metal compound and economical lithium carbonate as raw materials for synthesizing a lithium-containing transition-metal oxide. The obtained oxide has a high nickel content and high crystallinity. The use of lithium carbonate having a mean particle size of 6 mu m or less increases the effect of promoting crystal growth, suppressing the undesirable inclusion of nickel into the lithium site. Even when high-temperature baking is carried out, a disorder of a position interchange between a lithium ion and a nickel ion is suppressed. By switching the rotation direction, the evenness of the mixed materials can be enhanced. As a result, the reaction proceeds uniformly and the reactivity improves. When metal (M) contains aluminum or magnesium, the effect of enhancing thermal stability increases. When lithium-containing transition-metal oxide in which metal (M) is cobalt is included as an active material in a non-aqueous electrolyte secondary battery, polarization in the final stage of discharge is reduced.    DETAILED DESCRIPTION - A transition-metal compound containing nickel and metal (M) other than lithium and nickel are mixed in a molar ratio of a:(1-a) with lithium carbonate in a preset ratio. The temperature of the mixture is made to reach a preset temperature range while repeatedly raising and lowering the temperature. The transition-metal compound is then reacted with lithium carbonate at the preset temperature range, to obtain a lithium-containing transition-metal oxide of formula: Li(Lix(NiaM1-a)1-x)O2.    x=0 or more, preferably 0-0.3;and    a=greater than 0, preferably 0.5-0.8.    DESCRIPTION OF DRAWING(S) - The drawing shows a schematic view of a device used for synthesizing a hydroxide by co-precipitation.    Pump (1)    Supply port (2)    Reaction vessel (3)    Product collection section (4)