We introduce the notion of nilpotent p.p. rings, and prove that the nilpotent p.p. condition is preserved over polynomial rings and skew polynomial rings.

Let R be a ring with an endomorphism  and an -derivation . Antoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. In this paper we introduce and investigate the notion of nil-( ,  )-compatible rings. The class of nil-( ,  )-compatible rings are extended through various ring extensions and many classes of nil-( ,  )-compatible rings are constructed. We also prove that, if R is nil--compatible and nil-Armendariz ring of power series type with nil  R  nilpotent, then nil(R[[x;  ]])  nil(R)[[x;  ]]. We show that, if R is a nil-Armendariz ring of power series type, with nil  R  nilpotent and nil-( ,  )-compatible ring, then nil  R  x;  ,     nil  R   x;  ,   . As a consequence, several known results are unified and extended to the more general setting. Also examples are provided to illustrate our results.

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This manuscript introduces and investigates various types of mappings in the context of a $C^*$-algebra valued metric space. These include conditionally compatible, non-compatible, and faintly compatible mappings. Additionally, the manuscript establishes fixed-point results for certain combinations of these mappings. Specifically, it considers pairs of self-mappings that satisfy either a $(\phi-\psi)$-Kannan-type or Chatterjea-type contractive condition and proves common fixed point theorems for faintly compatible continuous or reciprocal continuous along with non-compatible pairs of such mappings. Also, non-compatibility is relaxed by using the $E.A.$ property and faintly compatible continuous property or reciprocal continuous property for pairs of self-mappings. To demonstrate the practicality of these results, the manuscript includes examples that illustrate their usefulness.

Using fixed point methods, we prove the stability of orthogonally ring homomorphism and orthogonally ring derivation in Banach algebras.

Here, the existence of fixed points for weakly compatible maps is studied. The results are new generalization of the results of [5]. Finally, we study the new common fixed point theorems.