In the paper I considered mr of polynomials over associative D-algebra with unit. Using the tensor notation allows to simplify the representation wrie polynomial. I considered questions related to esasy of polynomial of any power over polynomial of power 1. Such algsbra like the theorem on the uniqueness of the decomposition write me algebra essay the polynomial into a product of irreducible polynomials section the remainder theorem , p.
The theory of polynomials over non-commutative algebra is more difficult , p. In this paper, we attempted to advance a bit in this field.
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This statement is write me essy essay on the theorem 4. Opportunity to present a polynomial ewsay tensor eliminates the complexity and allows see important properties of polynomial. Using theorems considered in , Click proved alegbra theorem 6. I hope this is first step to study divisibility of polynomials. Alexandre Laugier was first reader of my paper.
I appreciate his helpful comments. I assume sum over alvebra s in expression like a s 0 xa s 1 Convention 2. Let A be free just click for source with finite or countable basis. Considering expansion of element of algebra A relative basis e we use the same root letter to denote this element write me algebra essay its coordinates.
To make text clearer we use separate color for index of element wrote algebra. It is very algbra to draw drite line between the module and the algebra.
Especially since sometimes in the process of constructing, we must first write me algebra essay that the set A is a module, and then we prove that this set is an algebra. Therefore, to write the element of the module, we will also use the convention 2. For instance, complex number is also called C-number, and quaternion is called H-number. Let Write me algebra essay rssay algebra of 3 3 matrices.
Let a be right zero divisor of D-algebra A. Let b be left zero divisor of D-algebra A. Then for any d A, essah is zero divisor of D-algebra A. There exists D-algebra where left zero divisor is not right zero source. Wssay A be free R-vector space which has Hamel basis e.
Let h L R;A;A be such map that 3. There exists D-algebra where left zero divisor is invertible from right. However, the matrix f is invertible write me algebra essay right. It is possible that this phenomenon is associated with the statement that a countable set has proper countable subset. Let map f L Esszy be defined by the equation 3. Let map g L R;A;A be defined by zlgebra equation write me algebra essay. According to the statement 3. According to the equation 3.
Let a be left zero divisor of D-algebra A. Article source zero element of right ideal Aa is left zero divisor of D-algebra A. According to the definition 3. If d is left zero divisor, then, according to read article write me algebra essay 3.
So to prove the theorem, we consider the case when d is not left mee divisor. Therefore, c is left zero divisor.
Let neither a A, write me algebra essay b A be left zero divisors of D-algebra A. Then their product ab is not left zero divisor of D-algebra A. If the product ab is a left zero divisor of D-algebra A, then, according to the theorem 3.
write me algebra essay Write me algebra essay
This contradiction proves the theorem. To see better the structure of the set of zero divisors write me algebra essay D-algebra A, we consider the following theorem. Let A be finite dimensional D-algebra. Let e be a basis of D-algebra A. Let C i kl be structural constants of D-algebra A relative write me algebra essay the basis e. Then coordinates a i of left zero divisor 3.
Since A-number a 0 is left zero divisor, then according to the definition 3. Therefore, coordinatesofa-numbersaand b satisfy to the system of equations 3. Let p k x be monomial of power k article source D-algebra A. We prove the theorem by induction over write me algebra essay n of monomial. We get the statement since monomial p 0 x is constant. Last factor of monomial p k x is either a k A, or has form x l, l 1.
Factor preceding a k has form x l, l aglebra. We can represent this factor as x l 1 x.
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Therefore, we proved the statement. We denote A k [x] Abelian group generated by the set of monomials of alebra k. Element p k x of Abelian group A k [x] is called homogeneous polynomial of power k.
According to the definition 4. Abelian group A k [x] is A-module. For any tensor a b A A, there exists transformation of monomial 4. Therefore, Abelian group A k [x] is A-module. You can see similar definition of monomial over Banach algebra in the section  For instance, For polynomial, we will use notation Theorem 4. The map defined by the equation is polylinear map.
Therefore, the map f is polylinear map. There exists linear map a learn more here algegra The theorem follows from theorems  An element p x of A-module A[x] is called polynomial over D-algebra A. Therefore, we can present polynomial of power n in the following click here Definition wssay. On the same page, Lang proves the existence of click here sum of modules.
Operations over Mee Definition 5. According to the corollary 4. According to the definition 5. Then the polynomial p x can be represented using one of the following forms 5. Based on the statement and the theorem 4. According to the remark 4. The theorem follows from the theorem 5. In this section, it is unimportant for us whether a polynomial p x is zero divisor.
We consider in the section 6 the question about zero divisors of A-algebra A[x]. In the proof of the theorems in this section, we, without loss of generality, assume that the considered polynomials are not zero divisors. Since product of homogeneous polynomials is bilinear map, then the definition 5. We prove the theorem by induction over n, m. Following lemma follows from the definition 5.
Lemma The write me algebra essay 5. Therefore, we proved ne following lemma. Since product of algegra is bilinear write me algebra essay, then 5. Theorem A-module A[x] equipped wrige the write me algebra essay 5. The theorem follows from definitions  Consider the linear equation 13 6.
According to the theorem we can write the equation 6. The tensor a A 2 write me algebra essay called nonsingular tensor if this tensor satisfies to condition 6. Let F be a field. Let A be finite dimensional F-algebra. For the linear equation write me algebra essay. Additional research is required before we can state theorems of the section 6 for F-algebra with countable basis.
It is difficult to say whether the condition 6.
Автор: Gotaur Voodookazahn