Lambert's suicide, on May 6, was just two days shy of the 11th anniversary of his mother's death.
Plato struggled with drugs and alcohol problems from an early age.
Her death in 1999, at the age of 34, was the result of an accidental prescription drug overdose of Loritab, a painkiller, and Valium, authorities previously told the New York Times.
She was on her way back to Los Angeles with fiance Robert Menchaca after an appearance on the "Howard Stern Show," where she insisted she had "been clean and sober for over a decade" at the time of her death.
Lambert was just 14 when his mother died.
She also told the mag's Web site that Lambert had been experimenting with alcohol and drugs.
"He was talking so sweet to me on the phone that morning," she told the Web site. "But you never can tell what's in their mind."
Richardson said there would be no memorial.
"Our lives have just been wrapped around him," she said. "It's just devastating."
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