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We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; Z3 ⊕ Z3; Z3 ⊕ B where B is a Boolean ring; local ring with nil Jacobson radical; M2(Z2) or M2(Z3); or the ring of a Morita context with zero pairings where the underlying rings are Z2 or Z3.

Let R be an associative ring with unity. An element a ∈ R is said to be r-clean if a = e + r, where e is an idempotent and r is a von Neumann regular element in R. If every element of R is r-clean, then R is called an r-clean ring. In this paper, we investigate the conditions under which the group ring RG is r-clean. We show that if R is a ring and G is a locally finite p-group with p ∈ J(R), then the group ring RG is r-clean if and only if R is r-clean.

A ring R is (weakly) nil clean provided that every element in R is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let R be abelian, and let n ∈ ℕ. We prove that M n (R) is nil clean if and only if R/J(R) is Boolean and M n (J(R)) is nil. Furthermore, we prove that R is weakly nil clean if and only if R is periodic; R/J(R) is ℤ3, B or ℤ3 ⊕ B where B is a Boolean ring, and that M n (R) is weakly nil clean if and only if M n (R) is nil clean for all n ≥ 2.

A ring R is unit nil-clean if, for any $$a\in R$$ , there exists a unit $$u\in R$$ , such that ua is the sum of an idempotent and a nilpotent. In this paper, we completely describe the structure of unit nil-clean rings. We thereby provide a large class of rings over which every square matrix is equivalent to the sum of idempotent and nilpotent matrices. Furthermore, the uniqueness is determined by the abelian property.

A ring R is a Garcia ring provided that the product of two regular elements is unit-regular. We prove that every regular element in a Garcia ring R is the sum/difference of an idempotent and a unit. Furthermore, we prove that every regular element in a weak Garcia ring is the sum of an idempotent and a one-sided unit. These extend several known theorems on (one-sided) unit-regular rings to wider classes of rings with sum summand property.

The unit sum number, u(R), of a ring R is the least k such that every element is the sum of k units; if there is no such k then u(R) is ω or ∞ depending on whether the units generate R additively or not. Here we introduce a finer classification for the unit sum number of a ring and in this new classification we completely determine the unit sum number of the ring of integers of a quadratic field. Further we obtain some results on cubic complex fields which one can decide whether the unit sum number is ω or ∞. Then we present some examples showing that all possibilities can occur.

The unit sum number u(R) of a ring R is the least k such that every element is the sum of k units; if there is no such k then u(R) is ω or ∞ depending whether the units generate R additively or not. If RM is a left R-module, then the unit sum number of M is defined to be the unit sum number of the endomorphism ring of M. Here we show that if R is a ring such that R/J(R) is semisimple and $\Z_{2}$ is not a factor of R/J(R) and if P is a projective R-module such that JP ≪ P, (JP small in P), then u(P)= 2. As a result we can see that if P is a projective module over a perfect ring then u(P)=2.