On Left T-Nilpotent Rings

It is shown that any ring being a sum of two left T-nilpotent subrings is left T-nilpotent. The paper contains the description of all the semigroups S such that an S-graded ring \(R=\bigoplus _{s\in S}A_s\) has the property that the left T-nilpotency of all subrings among the subgroups \(A_s\) of the additive group of R implies the left T-nilpotency of R. Furthermore, this result is extended to rings R being S-sums.