This thesis studies nil-clean rings, in which every element can be expressed as the sum of an idempotent and a nilpotent element. Special attention is given to strongly nil-clean rings, characterized by the property that the quotient modulo the Jacobson radical is Boolean and the radical itself is a nilideal, as well as to uniquely nil-clean rings, where every element has exactly one nil-clean decomposition. We also examine matrix rings and prove that $M_n(F)$ is nil-clean if and only if the field $F$ satisfies $F \cong \mathbb{F}_2$. In the final part, we analyze the probability that a randomly chosen element of a finite commutative ring is nil-clean, and show that in $\mathbb{Z}_{p^k}$ the proportion of nil-clean elements equals $\frac{2}{p}$. It follows that the rings $\mathbb{Z}_{2^k}$ are always nil-clean, while for odd primes $p$ this property does not hold. The results demonstrate that the theory of nil-clean rings is closely related to Boolean structures and radicals.
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