t-semicommutative rings

In this paper, we study the semicommutative properties of noncommutative rings in a more general setting. Specifically, we study semicommutative properties of rings determined by a tripotent. An example is provided to demonstrate that a right [Formula: see text]-semicommutative ring is not necessarily right [Formula: see text]-semicommutative. Furthermore, we restrict the right [Formula: see text]-semicommutative properties to nilpotent elements, which we show strictly contains the class of right [Formula: see text]-semicommutative rings and right [Formula: see text]-semicommutative rings. As a result, some well-known results on semicommutative rings in D. D. Anderson and V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra 27(6) (1999) 2847–2852; H. Kose, B. Ungor and A. Harmanci, Reversible ring property via idempotent elements, Georgian Math. J. 29(6) (2022) 851–862; H. Kose, B. Ungor and A. Harmanci, Semicommutativity of rings by the way of idempotents, Filomat 33(11) (2019) 3497–3508; G. Marks, Reversible and symmetric rings, J. Pure Appl. Algebra 174(3) (2002) 311–318; F. Meng and J. Wei, [Formula: see text]-symmetric rings, Comm. Contemp. Math. 20(3) (2018) 1750039 are unified and generalized.