Some conditions under which Jordan derivations are zero

In the current article, we obtain the following results: Let A be an algebra and P be a semi-prime ideal of A. Suppose that d:A→(A/P) is a Jordan derivation such that dim{d(a)|a∈A}≤1. If d(P)={0}, then d is zero. As an application of this result, we prove that if A is an algebra such that ⋂P∈Σ(A)P={0}, where Σ(A) denotes the set of all semi-prime ideals of A, and further each semi-prime ideal of A is of codimension 1, then A is commutative.