In this article, we introduce the concept of graphs associated with commutative UP-algebra, which we say is a UP-graph whose vertices are the elements of commutative UP-algebra and whose edges are the association of two vertices, that is two elements from commutative UP-algebra. We also define a graph of equivalence classes of a commutative UP-algebra and prove some related results based on the algebraic properties of the graph. We show that two graphs are the same and complete bipartite if they are formed by equivalence classes of UP-algebra and the graph folding of commutative UP-algebra. An algorithm for checking whether a given set is a UP-algebra or not has also been given.

Let $\Lambda$ denote a commutative ring with unity and $\mathfrak{D}\left(\Lambda \right)$ denote a collection of all annihilating ideals from $\Lambda$. An annihilator intersection graph of $\Lambda$ is represented by the notation $\mathcal{AIG}\left(\Lambda \right)$. This graph is not directed in nature, where the vertex set is represented by $\mathfrak{D}{\left(\Lambda \right)}^{*}$. There is a connection in the form of an edge between two distinct vertices $\varsigma$ and $\varrho$ in $\mathcal{AIG}\left(\Lambda \right)$ iff $Ann\left(\varsigma \varrho \right)\ne Ann\left(\varsigma \right)\cap Ann\left(\varrho \right)$. In this work, we begin by categorizing commutative rings $\Lambda$, which are finite in structure, so that $\mathcal{AIG}\left(\Lambda \right)$ forms a star graph/2-outerplanar graph, and we identify the inner vertex number of $\mathcal{AIG}\left(\Lambda \right)$. In addition, a classification of the finite rings where the genus of $\mathcal{AIG}\left(\Lambda \right)$ is 2, meaning $\mathcal{AIG}\left(\Lambda \right)$ is a double-toroidal graph, is also investigated. Further, we determine $\Lambda$, having a crosscap 1 of $\mathcal{AIG}\left(\Lambda \right),$ indicating that $\mathcal{AIG}\left(\Lambda \right)$ is a projective plane. Finally, we examine the domination number for the annihilator intersection graph and demonstrate that it is at maximum, two.

${\mathfrak{R}}_{\mathfrak{c}}$ represents commutative rings that have unity elements. The collection of all zero-divisor elements in ${\mathfrak{R}}_{\mathfrak{c}}$ are represented by $D\left({\mathfrak{R}}_{\mathfrak{c}}\right)$. We denote an extended zero-divisor graph by the notation ${\daleth }^{\prime }\left({\mathfrak{R}}_{\mathfrak{c}}\right)$ of ${\mathfrak{R}}_{\mathfrak{c}}.$ This graph has a set of vertices in $D{\left({\mathfrak{R}}_{\mathfrak{c}}\right)}^{*}$. The graph ${\daleth }^{\prime }\left({\mathfrak{R}}_{\mathfrak{c}}\right)$ illustrates interactions among the zero-divisor elements of ${\mathfrak{R}}_{\mathfrak{c}}.$ Specifically, two different vertices u and y are connected in ${\daleth }^{\prime }\left({\mathfrak{R}}_{\mathfrak{c}}\right)$ iff $u{\mathfrak{R}}_{\mathfrak{c}}\cap Ann\left(y\right)$ is non-null or $y{\mathfrak{R}}_{\mathfrak{c}}\cap Ann\left(u\right)$ is non-null. The main idea for this work is to systematically analyze the ring ${\mathfrak{R}}_{\mathfrak{c}}$ which is finite for the unique aspect of their extended zero-divisor graph. This study particularly focuses on instances where the extended zero-divisor graph has a genus or crosscap of two. Furthermore, this work aims to thoroughly characterize finite ring ${\mathfrak{R}}_{\mathfrak{c}}$ wherein the extended zero-divisor graph ${\daleth }^{\prime }\left({\mathfrak{R}}_{\mathfrak{c}}\right)$ has an outerplanarity index of two. Finally, we determine the book thickness of ${\daleth }^{\prime }\left({\mathfrak{R}}_{\mathfrak{c}}\right)$ for genus at most one.

Let $\mathcal{R}$ be a ring. An additive map $x\mapsto x^*$ of $\mathcal{R}$ into itself is called an involution if (i) $(xy)^*=y^*x^*$ and (ii) $(x^*)^*=x$ hold for all $x,y\in \mathcal{R}$. In this paper, we study the effect of involution $"*"$ on prime rings that satisfying certain differential identities. The identities considered in this manuscript are new and interesting. As the applications, many known theorems can be either generalized or deduced. In particular, a classical theorem due to Herstein [A note on derivation II, Canad. Math. Bull., 1979] is deduced.

The paper investigates nonadditive mappings \(\Omega: \Re \to \Re\) on a prime ring \(\Re\) with involution \( * \), characterized by satisfying one of the following conditions: \begin{enumerate} \item \( [\Omega(u), \Omega(v)]_\bullet = [u, v]_\bullet \) for all \( u, v \in \Re \). \item \( [\Omega(u), v]_\bullet = [u, \Omega(v)]_\bullet \) for all \( u, v \in \Re \). \item \( \Omega(u \bullet v) = \Omega(u) \bullet v \) for all \( u, v \in \Re \). \end{enumerate} Furthermore, the paper characterizes generalized bi-skew Jordan derivations within prime \( * \)-rings and examines the implications of these results in the context of various operator algebras.