Infinite easier Waring constants for commutative rings

Suppose n ⩾ 2 . We show that there is no integer v ⩾ 1 such that for all commutative rings R with identity, every element of the subring J ( 2 n , R ) of R generated by 2 n -th powers can be written in the form ± f 1 2 n ± ⋯ ± f v 2 n for some f 1 , … , f v ∈ R and some choice of signs.