We obtain a complete structural characterization of Cohn–Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of finitely generated projective modules is atomic and cancelative. We introduce the nonunital generalizations of graded analogs of noetherian and artinian rings, graded locally noetherian and graded locally artinian rings, and characterize graded locally noetherian and graded locally artinian Leavitt path algebras without any restriction on the cardinality of the graph. As a consequence, we relax the assumptions of the Abrams–Aranda–Perera–Siles characterization of locally noetherian and locally artinian Leavitt path algebras.

It is shown that the complex semigroup algebra of a free monoid of rank at least two is *-primitive, where * denotes the involution on the algebra induced by word-reversal on the monoid.

We consider the isomorphism problem for partial group rings $R_{\hbox{\scriptsize\it par}}G$ and show that, in the modular case, if $\textit{char}(R)\,{=}\,p$ and $R_{\hbox{\scriptsize\it par}}G_1\,{\cong}\, R_{\hbox{\scriptsize\it par}}G_2$ then the corresponding group rings of the Sylow $p$-subgroups are isomorphic. We use this to prove that finite abelian groups having isomorphic modular partial group algebras are isomorphic. Moreover, in the integral case, we show that the isomorphism of partial group rings of finite groups $G_1$ and $G_2$ implies $\Z G_1\,{\cong}\, \Z G_2$.