States Idempotent Lifting Across Nilpotent Ideals: Let R be a unital ring, let I R be a two-sided nilpotent ideal, and let n N. If e M_n(R/I) is an idempotent matrix, then there exists an idempotent matrix e M_n(R) whose image under the quotient map M_n(R) M_n(R/I) is e.