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تسجيل مستخدم جديدLinear Map of $D$-Algebra
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تأليف
Aleks Kleyn
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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 point of view I started the book from consideration of Cartesian product of representations. Polymorphism of representations is a map of Cartesian product of representations which is a morphism of representations with respect to each separate independent variable. Reduced morphism of representations allows us to simplify the study of morphisms of representations. However a representation has to satisfy specific requirements for existence of reduced polymomorphism of representations. It is possible that Abelian group is only $Omega$-algebra, such that representation in this algebra admits polymorphism of representations. However, today, this statement has not been proved. Multiplicative $Omega$-group is $Omega$-algebra in which product is defined. The definition of tensor product of representations of Abelian multiplicative $Omega$-group is based on properties of reduced polymorphism of representations of Abelian multiplicative $Omega$-group. Since an algebra is a module in which the product is defined, then we can use this theory to study linear map of algebra. For instance, we can study the set of linear transformations of $D$-algebra $A$ as representation of algebra $Aotimes A$ in algebra $A$.
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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
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equation has roots $x_1$, $x_2$, $x_2=-x_1$. I considered different forms of the Vietes theorem and a possibility to apply the method of completing the square. In quaternion algebra, there exists quadratic equation which either has $1$ root, or has no roots.
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 $Om
ega$-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.
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uations in quaternion algebra. In order to study homogeneous system of linear differential equations, I considered vector space over division $D$-algebra, solving of linear equations over division $D$-algebra and the theory of eigenvalues in non commutative division $D$-algebra.
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