Mathematics Introduction to Algebra Starting with Modules

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Mathematics
The first step to abstract algebra is to introduce a linear space on the body and then generalize the concept of linear space to a module on the ring. The following is the flow of introduction in the order of body → ring → group. The concrete table of contents is as follows : Chapter 1 Modules on the body (also known as linear space or vector space) Chapter 2 Modules on a variable polynomial ring Chapter 3 Modules on a ring Chapter 4 Rational integer ring Chapter 5 Calculation theory of modules on a variable polynomial ring Chapter 7 Application of Module theory Chapter 7 From commutative group to noncommutative group Chapter 1 introduces a linear space of body coefficients and derives various dimensional formulas using short complete series. Chapter 2 reviews the theory of eigenvalues of matrices as a theory of modules on a variable polynomial ring. Chapter 3 removes the axiom of division from the axiom of the body and introduces the axiom of the ring by removing the commutative law of multiplication and then introduces the axiom of the ring by introducing the axiom of group theory. Chapter 3 reviews the theory of eigenvalues of matrices as a theory of modules on a variable polynomial ring. Chapter 3 removes the axiom of division from the axiom of the body and introduces the axiom of the ring by removing the axiom of the formula, and then introduces the axiom of group theory to define modules and modules on the ring. Chapter 4 explains the strict treatment of properties of integers learned in high school and gives the Smith standard form. Chapter 5 explains that properties parallel to Chapter 4 hold for monovariable polynomial rings and introduces the axiom of the ring by removing the axiom of the formula, and then introduces the axiom of the group to define modules and modules on the ring. Chapter 4 explains the structural theorem of finite generating Abelian groups, the Jordan standard form, the Cayley-Hamilton theorem, and the modules of the Sylvester equation. Chapter 6 explains how to calculate the subgroup of finite Abelian groups. Up to this point, only the additive group (with the action of the ring) comes out. Chapter 6 explains the treatment of non-commutative Jordan standard form, the Cayley-Hamilton theorem, and the modules of the Sylvester equation. It also explains how to calculate the subgroup of the finite Abelian group. So far, only the additive group (with the action of the ring) comes out. Chapter 6 explains the treatment using the normal form of Jordan standard form, the Cayley-Hamilton theorem, and the modules of the Sylvester equation. It also explains how to calculate the of the finite Abelian group. So far, only the additive group (with the action of the ring) comes out. Chapter 7 introduces the quotient and