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Chen M. and et al. have solved an open problem related to rank equalities for the sum of finitely many idempotent matrices using the Gaussian elimination method in [Chen M. and et al., On the open problem related to rank equalities for the sum of finitely many idempotent matrices and its applications, The Scientific World Journal, 2014]. In this work, it is obtained a similar rank equality for the sum of finitely many involutive matices and derived some results from this equality, Chen M. and et al. have solved the open problem related to rank equalities for the sum of finitely many idempotent matrices by using the Gussian elimination method (Chen M. and et al. “On the open problem related to rank equalities for the sum of finitely many idempotent matrices and its applications”, The Scientific World Journal, 2014). In this work, it is obtained a similar rank equality for the sum of finitely many involutive matices and derived some results from this equality.

In this paper we call a ring R delta(r)-clean if every element is the sum of an idempotent and an element in delta(R-R) where delta(R-R) is the intersection of all essential maximal right ideals of R. If this representation is unique (and the elements commute) for every element we call the ring uniquely (strongly) delta(r)-clean. Various basic characterizations and properties of these rings are proved, and many extensions are investigated and many examples are given. In particular, we see that the class of delta(r)-clean rings lies between the class of uniquely clean rings and the class of exchange rings, and the class of uniquely strongly delta(r)-clean rings is a subclass of the class of uniquely strongly clean rings. We prove that R is delta(r)-clean if and only if R/delta(r)(R-R) is Boolean and R/Soc(R-R) is clean where Soc(R-R) is the right socle of R.