Idempotent-prime ideals and Peirce decompositions in noncommutative rings

We study idempotent-prime ideals in associative rings with identity. A proper two-sided ideal $ P $ of $ R $ is idempotent-prime if $ aR(1-a)\subseteq P $ implies $ a\in P $ or $ 1-a\in P $, for every $ a\in R $. For $ A = R/P $, the main quotient criterion shows that this condition is equivalent to the absence of nontrivial idempotents $ e\in A $ satisfying $ eA(1-e) = 0 $. Thus, idempotent-primeness rules out nontrivial triangular Peirce decompositions in the quotient. In the commutative case, it is equivalent to connectedness of $ R/P $, or equivalently of $ V(P)\subseteq \text{Spec } R $. After developing the quotient criterion and its Peirce consequences, we treat the functorial behavior, comaximal ideals, Jacobson-radical lifting, local and semisimple Artinian quotients, torsion-theoretic and avoidance interpretations, finite direct products, matrix rings, triangular matrix rings, topological descriptions, and graph-theoretic criteria.