Some aspects of rings whose elements satisfy a special property

Throughout, rings are associative with identity 1 ≠ 0. The main focus of this thesis is on the study of rings whose elements are sums of certain special elements and, in this case, these elements are from the sets of nilpotents, units, idempotents, involutions, and tripotents. In Chapter 1, we provide some basic definitions and results in ring theory which are needed for this thesis. In Chapter 2, we study rings whose elements are sums of a nilpotent and an idempotent (i.e., nil-clean rings). The motivation is the open question, raised by Breaz et al., whether the ring of linear transformations of a countable dimensional vector space over F₂ is a nil-clean ring. In Section 2:1, we first prove that for a semisimple module M over a ring R with R ∕J(R) Boolean, every endomorphism of M is a sum of an idempotent endomorphism and a locally nilpotent endomorphism. As a consequence, it is proved that, for a vector space V over a division ring D, every linear transformation of V is a sum of an idempotent linear transformation and a locally nilpotent linear transformation if and only if D ≅ F₂. In Section 2:2, we study nil ⁎-clean rings with emphasis on matrix rings. We show that a ⁎-ring R is a nil ⁎-clean ring if and only if J(R) is nil and R ∕J(R) is a nil ⁎-clean ring. Particularly, it is shown that for a 2-primal ⁎-ring R, with involution ⁎ given by (aᵢj)* = (a*ᵢj)ᵀ , Mₙ(R) is a nil ⁎-clean ring if and only if J(R) is nil, R ∕J(R) is Boolean, a* ─ a ∈ J(R) for all a ∈ R and Mₙ(F₂) is nil ⁎-clean. Notice that the structure of nil-clean rings is unknown. We give a description of nil-clean rings with nilpotency index at most 2 in Section 2:3. In Chapter 3 we first give a review of known results regarding rings whose elements are sums of nilpotents, idempotents, or tripotents. In Section 3:2 we determine the structure for a more general class of rings i.e., rings in which every element is a sum of a nilpotent and three tripotents that commute with one another and, in the last section, we discuss when a group ring has such property. In Chapter 4, we focus on decomposing a matrix as a sum of certain special matrices. In Section 4:1 we give the necessary and sufficient conditions for an n x n matrix over an integral domain to be a sum of involutions and, respectively, a sum of tripotents. We completely determine the integral domains over which every n x n matrix is a sum of involutions and, respectively, a sum of tripotents. We also show that every n x n matrix over an integral domain R is a sum of two tripotents if and only if R ≅ Fₚ where p = 2, 3, or 5. The last chapter of the thesis is about rings whose elements are left annihilator-stable. An element a in a ring R is left annihilator-stable (or left AS) if, whenever Ra + l(b) = R with b ∈ R, then a ─ u ∈ l(b) for some unit u in R, and the ring R is a left AS ring if each of its elements is left AS. In this chapter, we show that the left AS elements in a ring form a multiplicatively closed set, giving an affirmative answer to a question of Nicholson [66]. This result is further used to obtain a necessary and sufficient condition for a formal triangular matrix ring to be left AS. As an application, we provide examples of left AS rings R over which the triangular matrix rings Tₙ(R) are not left AS for all n ≥ 2. These examples give a negative answer to another question of Nicholson [66] whether R ∕J(R) being left AS implies that R is left AS