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The common in ring, module and algebra is that they are Abelian group with respect to addition. This property is enough to study integration. I treat integral of measurable map into normed Abelian $Omega$-group. Theory of integration of maps into $Omega$-group has a lot of common with theory of integration of functions of real variable. However I had to change some statements, since they implicitly assume either compactness of range or total order in $Omega$-group.
Since sum which is not necessarily commutative is defined in Omega-algebra A, then Omega-algebra A is called Omega-group. I also considered representation of Omega-group. Norm defined in Omega-group allows us to consider continuity of operations and continuity of representation.
Module is effective representation of ring in Abelian group. Linear map of module over commutative ring is morphism of corresponding representation. This definition is the main subject of the book. To consider this definition from more general poin
Let $A$ be Banach algebra over commutative ring $D$. The map $f:Arightarrow A $ is called differentiable in the Gateaux sense, if $$f(x+a)-f(x)=partial f(x)circ a+o(a)$$ where the Gateaux derivative $partial f(x)$ of map $f$ is linear map of incremen
In this paper, we formulate and prove Wendroffs inequalities on time scales. Next, we deduct some of Pachpattes inequalities.
Let $A$, $B$ be Banach $D$-algebras. The map $f:Arightarrow B$ is called differentiable on the set $Usubset A$, if at every point $xin U$ the increment of map $f$ can be represented as $$f(x+dx)-f(x) =frac{d f(x)}{d x}circ dx +o(dx)$$ where $$frac{d