In this paper, the author studies uniquely clean (uc) and uniquely nil clean (unc) rings. He gets some equivalent conditions for a (uc) ring.
The author proves that a ring \(R\) is a (uc) ring if and only if any one of the following conditions holds.

(i)

\(R\) is an Abelian exchange ring whose factor rings modulo maximal ideals are 2-element fields.

(ii)

If \(J^*(R)\) denotes the intersection of all maximal ideals of \(R\) then \(R\) is an exchange ring such that \(R/J^*(R)\) is Boolean and idempotents can be lifted uniquely modulo \(J^*(R)\).

(iii)

\(R/J^*(R)\) is Boolean and \(R\) is strongly clean in the sense that every element of \(R\) is a sum of a central idempotent and a unit in \(R\).

The author proves that every (uc) ring is a subdirect product of a family of local rings whose factor rings modulo their Jacobson radicals are 2-element fields. Next the author considers the conditions under which a (uc) ring is a (unc) ring. He proves that a (uc) ring in which every prime ideal is maximal is (unc). (In this proof, \(P(R)\) is nil seems to be well known.) As a corollary the author deduces that a commutative ring \(R\) is (unc) if and only if it is (uc) with every prime ideal maximal. The author also proves that a ring \(R\) is (unc) if and only if \(R\) is Abelian \(\pi\)-regular which is Boolean modulo its Jacobson radical. The author concludes the paper with the following result. A ring \(R\) is (unc) if and only if it is Abelian with its center \(C(R)\) (unc) and \(R=C(R)+N(R)\) where the set \(N(R)\) of all nilpotent elements in \(R\) is an ideal of \(R\).
The author has been very casual in giving references. At several places while quoting Theorem 2.1 he says it is Theorem 2.5. In proving Theorem 2.5, instead of saying “if” and “only if”, the author says “1 implies 2” and “2 implies 1”.