We present a new type of mappings called centrally extended α‐homoderivations of a ring ℜ (i.e., a map H from ℜ into ℜ which satisfies H(x + y) − H(x) − H(y) ∈ Z(ℜ) and H(xy) − H(x)H(y) − H(x)α(y) − α(x)H(y) ∈ Z(ℜ) for any x, y ∈ ℜ) where α is a mapping of ℜ and discuss the relationship between these mappings and other related mappings. Also, we study their effect on the center of a ring, and we give several outcomes on the commutativity of the ring.

This paper studies the effect of n ‐antihomoderivations on τ ‐prime rings. The first part establishes commutativity criteria by imposing natural commutator, centralizing, Lie‐type, and Jordan‐type identities on a nonzero τ ‐ideal. The second part investigates when centralizing conditions on the action of the n ‐antihomoderivation on τ ‐symmetric elements ensure that the ring satisfies the standard identity S ₄ . Thus, the paper contains two related but distinct contributions: commutativity results in the first part and an S ₄ result in the second part.