Lower bound for the Erdos-Burgess constant of finite commutative rings

Let R be a finite commutative unitary ring. An idempotent in R is an element e is an element of R with e(2) = e. The Erdos-Burgess constant associated with the ring R is the smallest positive integer l such that for any given l elements (repetitions are allowed) of R, say a(1), ..., a(l) is an element of R, there must exist a nonempty subset a(1), ..., a(l) is an element of R l) with J subset of {1, 2, ..., l} with Pi(j is an element of J )a(j) being an idempotent. In this paper, we give a lower bound of the Erdos-Burgess constant in a finite commutative unitary ring in terms of all its maximal ideals, and prove that the lower bound is attained in some cases. The result unifies some recently obtained theorems on this invariant.