In this article, we introduce the notion of the uniquely $I$-clean ring and show that, if $R$ is a ring and $I$ is an ideal of $R$ then $R$ is uniquely $I$-clean if and only if ($R/ I$ is Boolean and idempotents lift uniquely modulo $I$) if and only if (for each $a\in R$ there exists a central idempotent $e\in R$ such that $e- a\in I$ and $I$ is idempotent-free). We examine when ideal extension is uniquely clean relative to an ideal. Also we obtain conditions on a ring $R$ and an ideal $I$ of $R$ under which uniquely $I$-clean rings coincide with uniquely clean rings. Further we prove that a ring $R$ is uniquely nil-clean if and only if ($N(R)$ is an ideal of $R$ and $R$ is uniquely $N(R)$-clean) if and only if $R$ is both uniquely clean and nil-clean if and only if ($R$ is an abelian exchange ring with $J(R)$ nil and every quasiregular element is uniquely clean). We also show that $R$ is a uniquely clean ring such that every prime ideal of $R$ is maximal if and only if $R$ is uniquely nil-clean ring and $N(R)= {\mathrm{Nil} }_{\ast } (R)$.

A $\ast $-ring $R$ is called (strongly) $\ast $-clean if every element of $R$ is the sum of a unit and a projection (that commute). Vaš [‘$\ast $-Clean rings; some clean and almost clean Baer $\ast $-rings and von Neumann algebras’, J. Algebra 324(12) (2010), 3388–3400] asked whether there exists a $\ast $-ring that is clean but not $\ast $-clean and whether a unit regular and $\ast $-regular ring is strongly $\ast $-clean. In this paper, we answer these two questions. We also give some characterisations related to $\ast $-regular rings.

It is well known that a ring R is an exchange ring iff, for any a ∈ R, a−e ∈ (a2−a)R for some e2 = e ∈ R iff, for any a ∈ R, a−e ∈ R(a2−a) for some e2 = e ∈ R. The paper is devoted to a study of the rings R satisfying the condition that for each a ∈ R, a−e ∈ (a2−a)R for a unique e2 = e ∈ R. This condition is not left–right symmetric. The uniquely clean rings discussed in (W. K. Nicholson and Y. Zhou, Rings in which elements are uniquely the sum of an idempotent and a unit, Glasgow Math. J. 46 (2004), 227–236) satisfy this condition. These rings are characterized as the semi-boolean rings with a restricted commutativity for idempotents, where a ring R is semi-boolean iff R/J(R) is boolean and idempotents lift modulo J(R) (or equivalently, R is an exchange ring for which any non-zero idempotent is not the sum of two units). Various basic properties of these rings are developed, and a number of illustrative examples are given.

An element $a$ in a ring $R$ with identity is called strongly clean if it is the sum of an idempotent and a unit that commute. And $a\in R$ is called strongly $\pi$-regular if both chains $aR\supseteq a^2R\supseteq\cdots$ and $Ra\supseteq Ra^2\supseteq\cdots$ terminate. A ring $R$ is called strongly clean (respectively, strongly $\pi$-regular) if every element of $R$ is strongly clean (respectively, strongly $\pi$-regular). Strongly $\pi$-regular elements of a ring are all strongly clean. Let $\sigma$ be an endomorphism of $R$. It is proved that for $\varSigma r_ix^i\in R[[x,\sigma]]$, if $r_0$ or $1-r_0$ is strongly $\pi$-regular in $R$, then $\varSigma r_ix^i$ is strongly clean in $R[[x,\sigma]]$. In particular, if $R$ is strongly $\pi$-regular, then $R[[x,\sigma]]$ is strongly clean. It is also proved that if $R$ is a strongly $\pi$-regular ring, then $R[x,\sigma]/(x^n)$ is strongly clean for all $n\ge1$ and that the group ring of a locally finite group over a strongly regular or commutative strongly $\pi$-regular ring is strongly clean.

A ring $R$ with identity is called strongly clean if every element of $R$ is the sum of an idempotent and a unit that commute. For a commutative local ring $R$, $n=3,4$, and $m, k, s \in {\mathbb N}$ it is proved that ${\mathbb M}_n(R)$ is strongly clean if and only if ${\mathbb M}_n(R[[x]])$ is strongly clean if and only if ${\mathbb M}_n(R[[x_1, x_2, \ldots, x_m]])$ is strongly clean if and only if $ {\mathbb M}_n(\frac{R[x]}{(x^{k})})$ is strongly clean if and only if $ {\mathbb M}_n(\dfrac{R[x_{1}, x_{2}, \ldots , x_{s}]}{(x^{n_1}_{1}, x^{n_{2}}_{2}, \ldots , x^{n_{s}}_{s})}) $ is strongly clean if and only if ${\mathbb M}_n(R \propto R)$ is strongly clean where $ R\propto R=\{\scriptsize(\begin{array}{@{}c@{\quad}c@{}} a& b \\ 0& a \end{array}): a, b \in R \}$ is the trivial extension of $R$. This extends a result of J. Chen, X. Yang and Y. Zhou [$\mathbf{5}$] from $n=2$ to 3 and 4.