Generalizing Semi-n-Potent Rings

The present article deals with the problem of characterizing a widely large class of associative and possibly non-commutative rings. So, we define and explore the class of rings R for which each element in R is a sum of a tripotent element from R and an element from the subring ∆(R) of R which commute with each other, calling them strongly ∆-tripotent rings, or shortly just SDT rings. Succeeding in obtaining a complete description of these rings R modulo their Jacobson radical J(R) as the direct product of a Boolean ring and a Yaqub ring, our results somewhat generalize those established by Ko¸san-Yildirim-Zhou in Can. Math. Bull. (2019). Specifically, it is proved that if a ring R is SDT, then the factor ring R/J(R) is always reduced and 6 lies in J(R). Even something more, as already noticed before, it is shown that the quotient R/J(R) is a tripotent ring, which means that each of its elements satisfies the cubic equation x3 = x. Furthermore, examining triangular matrix rings Tn(R), we succeeded to classify its structure rather completely in the case where R is a local ring and n≥ 3 by establishing a satisfactory necessary and sufficient condition in terms of the ring R and its sections, resp., divisions.