A new class of rings are introduced which every element in the ring sum involution and tripotent elements. This called t-clean is a generalization invo-clean subclass clean rings. Some properties this investigate. For an application graph theory, defined set vertices order pairs them element. The two adjacent if only elements zero or product zero. graphs connecting, has diameter one girth three.

‎Let $R$ be a ring‎ ‎with the Jacobson radical $J(R)$ and let $picolon Rto R/J(R)$ be‎ the canonical map‎. ‎Then $pi$ induces an order preserving group homomorphism‎ ‎$K_0picolon K_0(R)to K_0(R/J(R))$ and an‎ ‎affine continuous map $S(K_0pi)$ between the state space $St(R/J(R))$ and the‎ ‎state space $St(R).$‎ ‎In this paper‎, ‎we consider the natural affine map $S(K_0pi).$ We give a condition ...