In this paper, R is considered a completely primary finite ring and Z(R) is its subset of all zero divisors (including zero), forming a unique maximal ideal. We give a construction of R whose subset of zero divisors Z(R) satisfies the conditions (Z(R))5 = (0); (Z(R))4 ̸= (0) and determine the structures of the unit groups of R for all its characteristics.

Let \(R\) be a commutative completely primary finite ring with a unique maximal ideal \(Z(R)\) such that \((Z(R))^{5}=(0) ;(Z(R))^{4} \neq(0)\). Then \(R / Z(R) \cong G F\left(p^{r}\right)\) is a finite field of order \(p^{r}\). Let \(R_{0}=G R\left(p^{k r}, p^{k}\right)\) be a Galois ring of order \(p^{k r}\) and of characteristic \(p^{k}\) for some prime number \(p\) and positive integers \(k, r\) so that \(R=R_{0} \oplus U \bigoplus V \bigoplus W \bigoplus Y\), where \(U, V, W\) and \(Y\) are \(R_{0} / p R_{0}\) - spaces considered as \(R_{0}\) modules generated by \(e, f, g\) and \(h\) elements respectively. Then \(R\) is of characteristic \(p^{k}\) where \(1 \leq k \leq 5\). In this paper, we investigate and determine the structures of the unit groups of some classes of commutative completely primary finite ring \(R\) with \(p u_{i}=p^{\xi} v_{j}=p w_{k}=p y_{l}=0\), where \(\xi=2,3 ; 1 \leq i \leq e, 1 \leq j \leq f, 1 \leq k \leq g\), and \(1 \leq l \leq h\).

A group action on a set is a process of developing an algebraic structure through a relation defined by the permutations in the group and the elements of the set. The process suppresses most of the group properties, emphasizing the permutation aspect, so that the algebraic structure has a wider application among other algebras. Such structures not only reveal connections between different areas in Mathematics but also make use of results in one area to suggest conjectures and also prove results in a related area. The structure (G, X) is a transitive permutation group G acting on the set X. Investigations on the properties associated with various groups acting on various sets have formed a subject of recent study. A lot of investigations have been done on the action of the symmetric group S n on various sets, with regard to rank, suborbits and subdegrees. However, the action of the dihedral group has not been thoroughly worked on. This study aims at investigating the properties of suborbits of the dihedral group D n acting on ordered subsets of { } 1, 2, , X n =  . The action of D n on X [r] , the set of all ordered r-element subsets of X, has been shown to be transitive if and only if n = 3. The number of self-paired suborbits of D n acting on X [r] has been determined, amongst other properties. Some of the results have been used to determine graphical properties of associated suborbital graphs, which also reflect some group theoretic properties. It has also been proved that when G = D n acts on ordered adjacent vertices of G, the number of self-paired suborbits is n + 1 if n is odd and n + 2 if n is even. The study has also revealed a conjecture that gives a formula for computing the self-paired suborbits of the action of D n on its ordered adjacent vertices. Properties of suborbits are significant as they form a link between group theory and graph theory.

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Recent developments in mathematics have in a sense organized objects of study into categories, where properties of mathematical systems can be unified and simplified through presentation of diagrams with arrows. A category is an algebraic structure made up of a collection of objects linked together by morphisms. Category theory has been advanced as a more concrete foundation of mathematics as opposed to set-theoretic language. In this paper, we define a pseudo-category on the class of isotonic spaces on which the idempotent axiom of the Kuratowski closure operator is assumed.