Here we prove that, for a $2$-primal ring $R$, the Laurent series ring $R((x))$ is a clean ring if and only if $R$ is a semiregular ring with $J(R)$ nil. This disproves the claim in K. I. Sonin [‘Semiprime and semiperfect rings of Laurent series’, Math. Notes 60 (1996), 222–226] that the Laurent series ring over a clean ring is again clean. As an application of the result, it is shown that, for a $2$-primal ring $R$, $R((x))$ is semiperfect if and only if $R((x))$ is semiregular if and only if $R$ is semiperfect with $J(R)$ nil.

In 1977 Hartwig and Luh asked whether an element $a$ in a Dedekind-finite ring $R$ satisfying $aR= {a}^{2} R$ also satisfies $Ra= R{a}^{2} $. In this paper, we answer this question in the negative. We also prove that if $a$ is an element of a Dedekind-finite exchange ring $R$ and $aR= {a}^{2} R$, then $Ra= R{a}^{2} $. This gives an easier proof of Dischinger’s theorem that left strongly $\pi $-regular rings are right strongly $\pi $-regular, when it is already known that $R$ is an exchange ring.

A ring $R$ is called left morphic if, for every $a\in R$, $R/Ra\cong {\bf l}(a)$ where ${\bf l}(a)$ denotes the left annihilator of $a$ in $R$. Right morphic rings are defined analogously. In this paper, we investigate when the trivial extension $R\propto M$ of a ring $R$ and a bimodule $M$ over $R$ is (left) morphic. Several new families of (left) morphic rings are identified through the construction of trivial extensions. For example, it is shown here that if $R$ is strongly regular or semisimple, then $R\propto R$ is morphic; for an integer $n>1$, ${\mathbb Z}_n\propto {\mathbb Z}_n$ is morphic if and only if $n$ is a product of distinct prime numbers; if $R$ is a principal ideal domain with classical quotient ring $Q$, then the trivial extension $R\propto {Q}/{R}$ is morphic; for a bimodule $M$ over $\mathbb Z$, ${\mathbb Z}\propto M$ is morphic if and only if $M\cong{\mathbb Q}/{\mathbb Z}$. Thus, ${\mathbb Z}\propto {\mathbb Q}/{\mathbb Z}$ is a morphic ring which is not clean. This example settled two questions both in the negative raised by Nicholson and Sánchez Campos, and by Nicholson, respectively.

An associative ring with unity is called clean if every element is the sum of an idempotent and a unit; if this representation is unique for every element, we call the ring uniquely clean. These rings represent a natural generalization of the Boolean rings in that a ring is uniquely clean if and only if it is Boolean modulo the Jacobson radical and idempotents lift uniquely modulo the radical. We also show that every image of a uniquely clean ring is uniquely clean, and construct several noncommutative examples.

A ring $R$ satisfies the dual of the isomorphism theorem if $R/Ra\cong \mathtt{l}(a)$ for every element $a\in R.$ We call these rings left morphic, and say that $R$ is left P-morphic if, in addition, every left ideal is principal. In this paper we characterize the left and right P-morphic rings and show that they form a Morita invariant class. We also characterize the semiperfect left P-morphic rings as the finite direct products of matrix rings over left uniserial rings of finite composition length. J. Clark has an example of a commutative, uniserial ring with exactly one non-principal ideal. We show that Clark's example is (left) morphic and obtain a non-commutative analogue.